Problem of Apollonius

2191:, as described below (Figure 6). In the second case, both roots are identical, corresponding to a solution circle that transforms into itself under inversion. In this case, one of the given circles is itself a solution to the Apollonius problem, and the number of distinct solutions is reduced by one. The third case of complex conjugate radii does not correspond to a geometrically possible solution for Apollonius' problem, since a solution circle cannot have an imaginary radius; therefore, the number of solutions is reduced by two. Apollonius' problem cannot have seven solutions, although it may have any other number of solutions from zero to eight. 4616: 12205: 7389:. The general number of solutions for each of the ten types of Apollonius' problem is given in Table 1 above. However, special arrangements of the given elements may change the number of solutions. For illustration, Apollonius' problem has no solution if one circle separates the two (Figure 11); to touch both the solid given circles, the solution circle would have to cross the dashed given circle; but that it cannot do, if it is to touch the dashed circle tangentially. Conversely, if three given circles are all tangent at the same point, then 993: 20: 640: 4955: 4021: 4124: 4078: 3997:.) Inversion has the useful property that lines and circles are always transformed into lines and circles, and points are always transformed into points. Circles are generally transformed into other circles under inversion; however, if a circle passes through the center of the inversion circle, it is transformed into a straight line, and vice versa. Importantly, if a circle crosses the circle of inversion at right angles (intersects perpendicularly), it is left unchanged by the inversion; it is transformed into itself. 7378: 4545:, the three given circles and the solution circle can be resized in tandem while preserving their tangencies. Thus, the initial Apollonius problem is transformed into another problem that may be easier to solve. For example, the four circles can be resized so that one given circle is shrunk to a point; alternatively, two given circles can often be resized so that they are tangent to one another. Thirdly, given circles that intersect can be resized so that they become non-intersecting, after which the 28: 3804: 7157: 7364: 7341: 7318: 7295: 7272: 7249: 7226: 7203: 7180: 3823:. The basic strategy of inversive methods is to transform a given Apollonius problem into another Apollonius problem that is simpler to solve; the solutions to the original problem are found from the solutions of the transformed problem by undoing the transformation. Candidate transformations must change one Apollonius problem into another; therefore, they must transform the given points, circles and lines to other points, circles and lines, and no other shapes. 12192: 7828: 7459: 2805: 7855:. Once the solutions to the planar problem have been constructed, the corresponding solutions to the spherical problem can be determined by inverting the stereographic projection. Even more generally, one can consider the problem of four tangent curves that result from the intersections of an arbitrary quadratic surface and four planes, a problem first considered by 1090: 3782: 4603:

of the inverted problem must either be (1) a straight line parallel to the two given parallel lines and tangent to the transformed third given circle; or (2) a circle of constant radius that is tangent to the two given parallel lines and the transformed given circle. Re-inversion and adjusting the radii of all circles by Δ

4978:; these two points are the two possible intersections of two tangent lines to the two circles. Therefore, the three given circles have six centers of similarity, two for each distinct pair of given circles. Remarkably, these six points lie on four lines, three points on each line; moreover, each line corresponds to the 2516: 4049:

blue points lying on each green line are transformed into one another. Hence, the lines connecting these conjugate tangent points are invariant under the inversion; therefore, they must pass through the center of inversion, which is the radical center (green lines intersecting at the orange dot in Figure 6).

105:: any of the three given circles can be shrunk to zero radius (a point) or expanded to infinite radius (a line). Viète's approach, which uses simpler limiting cases to solve more complicated ones, is considered a plausible reconstruction of Apollonius' method. The method of van Roomen was simplified by 7105:

Points and lines may be viewed as special cases of circles; a point can be considered as a circle of infinitely small radius, and a line may be thought of an infinitely large circle whose center is also at infinity. From this perspective, the general Apollonius problem is that of constructing circles

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case, the center must lie on a line bisecting the angle at the three intersection points between the three given lines; hence, the center lies at the intersection point of two such angle bisectors. Since there are two such bisectors at every intersection point of the three given lines, there are four

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circle tangent at the same point is a solution; such Apollonius problems have an infinite number of solutions. If any of the given circles are identical, there is likewise an infinity of solutions. If only two given circles are identical, there are only two distinct given circles; the centers of the

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of the general problem. These limiting cases often have fewer solutions than the general problem; for example, the replacement of a given circle by a given point halves the number of solutions, since a point can be construed as an infinitesimal circle that is either internally or externally tangent.

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The extension of Apollonius' problem to three dimensions, namely, the problem of finding a fifth sphere that is tangent to four given spheres, can be solved by analogous methods. For example, the given and solution spheres can be resized so that one given sphere is shrunk to point while maintaining

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Apollonius problem is to construct one or more circles tangent to three given objects in a plane, which may be circles, points, or lines. This gives rise to ten types of Apollonius' problem, one corresponding to each combination of circles, lines and points, which may be labeled with three letters,

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that intersects each of the two touching circles in two places. Upon inversion, the touching circles become two parallel lines: Their only point of intersection is sent to infinity under inversion, so they cannot meet. The same inversion transforms the third circle into another circle. The solution

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in the radical circle leaves the given circles unchanged, but transforms the two conjugate pink solution circles into one another. Under the same inversion, the corresponding points of tangency of the two solution circles are transformed into one another; for illustration, in Figure 6, the two

183:

The general statement of Apollonius' problem is to construct one or more circles that are tangent to three given objects in a plane, where an object may be a line, a point or a circle of any size. These objects may be arranged in any way and may cross one another; however, they are usually taken to

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Solutions to Apollonius's problem generally occur in pairs; for each solution circle, there is a conjugate solution circle (Figure 6). One solution circle excludes the given circles that are enclosed by its conjugate solution, and vice versa. For example, in Figure 6, one solution circle

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of the solution circle, which cancels out. This second formulation of Apollonius' problem can be generalized to internally tangent solution circles (for which the center-center distance equals the difference of radii), by changing the corresponding differences of distances to sums of distances, so

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to the first three factors cannot have positive dimension. This proves that generically, there are eight solutions counted with multiplicity. Since it is possible to exhibit a configuration where the eight solutions are distinct, the generic configuration must have all eight solutions distinct.

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which is the homogenization of the usual equation of a circle in the affine plane. Therefore, studying circles in the above sense is nearly equivalent to studying circles in the conventional sense. The only difference is that the above sense permits degenerate circles which are the union of two

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doubted whether Apollonius' problem could be solved by straightedge and compass. Viète first solved some simple special cases of Apollonius' problem, such as finding a circle that passes through three given points which has only one solution if the points are distinct; he then built up to solving

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should be chosen consistently with that of the first hyperbola. An intersection of these two hyperbolas (if any) gives the center of a solution circle that has the chosen internal and external tangencies to the three given circles. The full set of solutions to Apollonius' problem can be found by

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and undoing the resizing transforms such a solution line into the desired solution circle of the original Apollonius problem. All eight general solutions can be obtained by shrinking and swelling the circles according to the differing internal and external tangencies of each solution; however,

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tangency. Inversion in this point reduces Apollonius' problem to finding a plane that is tangent to three given spheres. There are in general eight such planes, which become the solutions to the original problem by reversing the inversion and the resizing. This problem was first considered by

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An exhaustive enumeration of the number of solutions for all possible configurations of three given circles, points or lines was first undertaken by Muirhead in 1896, although earlier work had been done by Stoll and Study. However, Muirhead's work was incomplete; it was extended in 1974 and a

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enclose the inner concentric circle, but rather revolve like ball bearings in the annulus. In the second family (Figure 8), the solution circles enclose the inner concentric circle. There are generally four solutions for each family, yielding eight possible solutions, consistent with the

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tangency is one in which the two circles curve in the same way at their point of contact; the two circles lie on the same side of the tangent line, and one circle encloses the other. In this case, the distance between their centers equals the difference of their radii. As an illustration, in

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determines a unique conic, its vanishing locus. Conversely, every conic in the complex projective plane has an equation, and that equation is unique up to an overall scaling factor (because rescaling an equation does not change its vanishing locus). Therefore, the set of all conics may be

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The property of tangency is defined as follows. First, a point, line or circle is assumed to be tangent to itself; hence, if a given circle is already tangent to the other two given objects, it is counted as a solution to Apollonius' problem. Two distinct geometrical objects are said to

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If the three given circles are mutually tangent, Apollonius' problem has five solutions. Three solutions are the given circles themselves, since each is tangent to itself and to the other two given circles. The remaining two solutions (shown in red in Figure 12) correspond to the

1843:= −1). For example, in Figures 1 and 4, the pink solution is internally tangent to the medium-sized given circle on the right and externally tangent to the smallest and largest given circles on the left; if the given circles are ordered by radius, the signs for this solution are 4576:

transforms the two given circles into new circles, and the solution circle into a line. Therefore, the transformed solution is a line that is tangent to the two transformed given circles. There are four such solution lines, which may be constructed from the external and internal

3226: 224:, meaning "touching".) In practice, two distinct circles are tangent if they intersect at only one point; if they intersect at zero or two points, they are not tangent. The same holds true for a line and a circle. Two distinct lines cannot be tangent in the plane, although two 200:

if they have a point in common. By definition, a point is tangent to a circle or a line if it intersects them, that is, if it lies on them; thus, two distinct points cannot be tangent. If the angle between lines or circles at an intersection point is zero, they are said to be

2475: 7739:, of which the first two stanzas are reproduced below. The first stanza describes Soddy's circles, whereas the second stanza gives Descartes' theorem. In Soddy's poem, two circles are said to "kiss" if they are tangent, whereas the term "bend" refers to the curvature 5212: 3583: 1093:

Figure 4: Tangency between circles is preserved if their radii are changed by equal amounts. A pink solution circle must shrink or swell with an internally tangent circle (black circle on the right), while externally tangent circles (two black circles on left) do the

5558: 6946: 5274:, respectively; thus, all six points can be located, from which one pair of solution circles can be found. Repeating this procedure for the remaining three homothetic-center lines yields six more solutions, giving eight solutions in all. However, if a line 1805: 6177: 1646: 1490: 4029:(pink, upper left) encloses two given circles (black), but excludes a third; conversely, its conjugate solution (also pink, lower right) encloses that third given circle, but excludes the other two. The two conjugate solution circles are related by 10721:

Apollonii de Tactionibus, quae supersunt, ac maxime lemmata Pappi, in hos libros Graece nunc primum edita, e codicibus manuscriptis, cum Vietae librorum Apollonii restitutione, adjectis observationibus, computationibus, ac problematis Apolloniani

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Apollonius' problem can be framed as a system of three equations for the center and radius of the solution circle. Since the three given circles and any solution circle must lie in the same plane, their positions can be specified in terms of the

2800:{\displaystyle \left(X_{1}-X_{2}\mid X_{1}-X_{2}\right)=2\left(v_{1}-v_{2}\right)\left(w_{1}-w_{2}\right)+\left(\mathbf {c} _{1}-\mathbf {c} _{2}\right)\cdot \left(\mathbf {c} _{1}-\mathbf {c} _{2}\right)-\left(s_{1}r_{1}-s_{2}r_{2}\right)^{2}.} 5911:

with another line in the projective plane (possibly the line at infinity again), and the other is union of two lines in the projective plane, one through each of the two circular points. These are the limits of smooth circles as the radius

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lies on the radical axis of the two solution circles. The same argument can be applied to the other pairs of circles, so that three centers of similitude for the given three circles must lie on the radical axes of pairs of solution circles.

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To solve the remaining problems, Viète exploited the fact that the given circles and the solution circle may be re-sized in tandem while preserving their tangencies (Figure 4). If the solution-circle radius is changed by an amount

77:

has survived. Three given circles generically have eight different circles that are tangent to them (Figure 2), a pair of solutions for each way to divide the three given circles in two subsets (there are 4 ways to divide a set of

2176:, which represents the same solution circle. Therefore, Apollonius' problem has at most eight independent solutions (Figure 2). One way to avoid this double-counting is to consider only solution circles with non-negative radius. 552:)—followed a similar progressive approach. Hence, Viète's solution is considered to be a plausible reconstruction of Apollonius' solution, although other reconstructions have been published independently by three different authors. 7640: 4777:, two points must be identified that lie on it; but these points need not be the tangent points. Gergonne was able to identify two other points for each of the three lines. One of the two points has already been identified: the 7982:

in distances to at least three points. For example, a ship may seek to determine its position from the differences in arrival times of signals from three synchronized transmitters. Solutions to Apollonius' problem were used in

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Either Soddy circle, when taken together with the three given circles, produces a set of four circles that are mutually tangent at six points. The radii of these four circles are related by an equation known as

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In the generic problem with eight solution circles, The reciprocals of the radii of four of the solution circles sum to the same value as do the reciprocals of the radii of the other four solution circles

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Figure 1, the pink solution circle is internally tangent to the medium-sized given black circle on the right, whereas it is externally tangent to the smallest and largest given circles on the left.

109:, who showed that Apollonius' problem is equivalent to finding a position from the differences of its distances to three known points. This has applications in navigation and positioning systems such as 4012:; hence, solutions of the planar Apollonius problem also pertain to its counterpart on the sphere. Other inversive solutions to the planar problem are possible besides the common ones described below. 3777:{\displaystyle \left(X_{\mathrm {sol} }\mid X_{\mathrm {sol} }\right)=\left(X_{\mathrm {sol} }\mid X_{1}\right)=\left(X_{\mathrm {sol} }\mid X_{2}\right)=\left(X_{\mathrm {sol} }\mid X_{3}\right)=0} 2308: 445:

A rich repertoire of geometrical and algebraic methods have been developed to solve Apollonius' problem, which has been called "the most famous of all" geometry problems. The original approach of

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Problema Apolloniacum quo datis tribus circulis, quaeritur quartus eos contingens, antea a...Francisco Vieta...omnibus mathematicis...ad construendum propositum, jam vero per Belgam...constructum

5075: 6816:, and therefore there are eight solutions to the problem of Apollonius, counted with multiplicity. To prove that the intersection is generically finite, consider the incidence correspondence 5887: 12963: 5629:

consisting of those points which correspond to conics passing through the circular points. Substituting the circular points into the equation for a generic conic yields the two equations

5354: 949:(1687) refined van Roomen's solution, so that the solution-circle centers were located at the intersections of a line with a circle. Newton formulates Apollonius' problem as a problem in 4521:

and the two given circles into concentric circles, with the third given circle becoming another circle (in general). This follows because the system of circles is equivalent to a set of

1152:. Thus, as the solution circle swells, the internally tangent given circles must swell in tandem, whereas the externally tangent given circles must shrink, to maintain their tangencies. 6822: 3549:

Therefore, Apollonius' problem can be re-stated in Lie geometry as a problem of finding perpendicular vectors on the Lie quadric; specifically, the goal is to identify solution vectors

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more complicated special cases, in some cases by shrinking or swelling the given circles. According to the 4th-century report of Pappus, Apollonius' own book on this problem—entitled

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has been defined for brevity, with the subscript indicating whether the solution is externally or internally tangent. A simple trigonometric rearrangement yields the four solutions

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on the right-hand side. Subtracting one equation from another eliminates these quadratic terms; the remaining linear terms may be re-arranged to yield formulae for the coordinates

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identify a receiver's position from the differences in arrival times of signals from three fixed positions, which correspond to the differences in distances to those transmitters.

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can be applied. In all such cases, the solution of the original Apollonius problem is obtained from the solution of the transformed problem by undoing the resizing and inversion.

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inherent in the problem of Apollonius: for instance solution circles generically occur in pairs, with one solution enclosing the given circles that the other excludes (Figure 2).

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of a potential pair of solution circles. To show this, Gergonne considered lines through corresponding points of tangency on two of the given circles, e.g., the line defined by

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Figure 11: An Apollonius problem with no solutions. A solution circle (pink) must cross the dashed given circle (black) to touch both of the other given circles (also black).

293:. If the solution circle is externally tangent to all three given circles, the distances between the center of the solution circle and the centers of the given circles equal 12378: 2006: 1954: 7718:

of the three given circles. For every set of four mutually tangent circles, there is a second set of four mutually tangent circles that are tangent at the same six points.

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to determine the location of an artillery piece from the time a gunshot was heard at three different positions, and hyperbolic trilateration is the principle used by the

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By solving Apollonius' problem repeatedly to find the inscribed circle, the interstices between mutually tangential circles can be filled arbitrarily finely, forming an

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are known functions of the given circles and the choice of signs. Substitution of these formulae into one of the initial three equations gives a quadratic equation for

6084: 3442: 3311: 9305:(reprint of 1929 edition by Houghton Mifflin ed.). New York: Dover Publications. pp. 117–121 (Apollonius' problem), 121–128 (Casey's and Hart's theorems). 525:, who had urged van Roomen to work on Apollonius' problem in the first place, developed a method that used only compass and straightedge. Prior to Viète's solution, 606:, although their solutions were rather complex. Practical algebraic methods were developed in the late 18th and 19th centuries by several mathematicians, including 3896: 518:. Therefore, van Roomen's solution—which uses the intersection of two hyperbolas—did not determine if the problem satisfied the straightedge-and-compass property. 5774: 5748: 4513:

and that intersect the two given circles orthogonally. These two constructed circles intersect each other in two points. Inversion in one such intersection point

1086:. Viète solved all ten of these cases using only compass and straightedge constructions, and used the solutions of simpler cases to solve the more complex cases. 7808:

Apollonius' problem can be extended to construct all the circles that intersect three given circles at a precise angle θ, or at three specified crossing angles θ

3795:, simultaneously perpendicular vectors. This gives another way to calculate the maximum number of solutions and extend the theorem to higher-dimensional spaces. 10429:

Coaklay GW (1859–1860). "Analytical Solutions of the Ten Problems in the Tangencies of Circles; and also of the Fifteen Problems in the Tangencies of Spheres".

7820:; the ordinary Apollonius' problem corresponds to a special case in which the crossing angle is zero for all three given circles. Another generalization is the 4565: 12313: 7063:

are much easier to solve than the general case of three given circles. The two simplest cases are the problems of drawing a circle through three given points (

7485: 921:, characterizes hyperbolas, so the possible centers of the solution circle lie on a hyperbola. A second hyperbola can be drawn for the pair of given circles 4378:{\displaystyle \cos \theta ={\frac {d_{\mathrm {s} }^{2}+d_{\mathrm {non} }^{2}-d_{\mathrm {T} }^{2}}{2d_{\mathrm {s} }d_{\mathrm {non} }}}\equiv C_{\pm }.} 9185:

Coaklay GW (1860). "Analytical Solutions of the Ten Problems in the Tangencies of Circles; and also of the Fifteen Problems in the Tangencies of Spheres".

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on the right-hand sides of the equations may be chosen in eight possible ways, and each choice of signs gives up to two solutions, since the equation for

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Viète used this approach to shrink one of the given circles to a point, thus reducing the problem to a simpler, already solved case. He first solved the

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circles in the plane has special properties, which have been elucidated by Larmor (1891) and Lachlan (1893). Such a configuration is also the basis for

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or trilateration, which is the task of locating a position from differences in distances to three known points. For example, navigation systems such as

8992:(in German) (reprinted in 1973 by Georg Olms Verlag (Hildesheim) ed.). Göttingen: Königlichen Gesellschaft der Wissenschaften. pp. 399–400. 12943: 7406:

definitive enumeration, with 33 distinct cases, was published in 1983. Although solutions to Apollonius' problem generally occur in pairs related by

10817: 4398: 2104:. This might suggest (incorrectly) that there are up to sixteen solutions of Apollonius' problem. However, due to a symmetry of the equations, if ( 757:

whose foci are the centers of the given circles. To understand this, let the radii of the solution circle and the two given circles be denoted as

143:

Apollonius' problem has stimulated much further work. Generalizations to three dimensions—constructing a sphere tangent to four given spheres—and

8011:). It is also used to determine the position of calling animals (such as birds and whales), although Apollonius' problem does not pertain if the 4890:

of the solution circles, by definition (Figure 9). The relationship between pole points and their polar lines is reciprocal; if the pole of

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are different (i.e. the circles have opposite "orientations"), the circles are externally tangent; the distance between their centers equals the

3004:{\displaystyle \left(X_{1}-X_{2}\mid X_{1}-X_{2}\right)=\left(X_{1}\mid X_{1}\right)-2\left(X_{1}\mid X_{2}\right)+\left(X_{2}\mid X_{2}\right).} 4065:

between the two concentric circles. Therefore, they belong to two one-parameter families. In the first family (Figure 7), the solutions do

12065: 10799: 10330:(September 1808). "Sur le contact des sphères; sur la sphère tangente à quatre sphères données; sur le cercle tangent à trois cercles donnés". 4962:

in the three given circles (black) lie on the green lines connecting the tangent points. These lines may be constructed from the poles and the

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are the same (i.e. the circles have the same "orientation"), the circles are internally tangent; the distance between their centers equals the

1132:. He then derived a lemma for constructing the line perpendicular to an angle bisector that passes through a point, which he used to solve the 12281: 8609: 8028: 471: 4557:

In the first approach, the given circles are shrunk or swelled (appropriately to their tangency) until one given circle is shrunk to a point

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problem can be solved as follows. The center of the solution circle is equally distant from all three points, and therefore must lie on the

10537: 7851:) that are tangent to three given circles on the sphere. This spherical problem can be rendered into a corresponding planar problem using 5320:

are allowed and degenerate situations are counted with multiplicity. When this is done, there are always eight solutions to the problem.

3221:{\displaystyle -2\left(X_{1}\mid X_{2}\right)=\left|\mathbf {c} _{1}-\mathbf {c} _{2}\right|^{2}-\left(s_{1}r_{1}-s_{2}r_{2}\right)^{2}.} 11665: 12370: 555:

Several other geometrical solutions to Apollonius' problem were developed in the 19th century. The most notable solutions are those of

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from the common concentric center to the non-concentric circle (Figure 7). The solution circle can be determined from its radius

1835:, called signs, may equal ±1, and specify whether the desired solution circle should touch the corresponding given circle internally ( 12143: 10730: 2187:

roots. The first case corresponds to the usual situation; each pair of roots corresponds to a pair of solutions that are related by

1302:, respectively. The requirement that a solution circle must exactly touch each of the three given circles can be expressed as three 12493: 11446: 10982: 7462:

Figure 12: The two solutions (red) to Apollonius' problem with mutually tangent given circles (black), labeled by their curvatures.

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from its center to the common concentric center and the center of the non-concentric circle, respectively. The radius and distance

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problem (two lines and a point). This accounts for the first four cases of Apollonius' problem, those that do not involve circles.

510:(the problem of constructing a cube of twice the volume of a given cube) cannot be done using only a straightedge and compass, but 3827:

has this property and allows the center and radius of the inversion circle to be chosen judiciously. Other candidates include the

136:

to simplify the configuration of the given circles. These developments provide a geometrical setting for algebraic methods (using

12918: 12895: 12651: 12599: 10954: 8209: 7030: 5312:, can be used to solve Apollonius's problem. In this approach, the problem is reinterpreted as a statement about circles in the 603: 4081:

Figure 7: A solution circle (pink) in the first family lies between concentric given circles (black). Twice the solution radius

2470:{\displaystyle \left(X_{1}\mid X_{2}\right):=v_{1}w_{2}+v_{2}w_{1}+\mathbf {c} _{1}\cdot \mathbf {c} _{2}-s_{1}s_{2}r_{1}r_{2}.} 12503: 12241: 11990: 9745: 9731: 5059: 4229:, depending on whether the solution circle is internally or externally tangent to the non-concentric circle. Therefore, by the 4057:

If two of the three given circles do not intersect, a center of inversion can be chosen so that those two given circles become

251:

at that point, and they exclude one another. The distance between their centers equals the sum of their radii. By contrast, an

6603:, respectively, counted with multiplicity and with the circular points deducted. The rational function determines a morphism 838:, depending on whether these circles are chosen to be externally or internally tangent, respectively. Similarly, the distance 11720: 10608: 10159: 10135: 9310: 9153: 9124: 8919: 8667: 8348: 8320: 8287: 8258: 7824:

of the first extension, namely, to construct circles with three specified tangential distances from the three given circles.

5207:{\displaystyle {\overline {X_{3}A_{1}}}\cdot {\overline {X_{3}A_{2}}}={\overline {X_{3}B_{1}}}\cdot {\overline {X_{3}B_{2}}}} 4568:, which is the problem of finding a solution circle tangent to the two remaining given circles that passes through the point 1054:

as the set of points that have a given ratio of distances to two fixed points. (As an aside, this definition is the basis of

487: 247:

tangency is one where the two circles bend away from each other at their point of contact; they lie on opposite sides of the

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of the three circles. For illustration, the orange circle in Figure 6 crosses the black given circles at right angles.

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Figure 8: A solution circle (pink) in the second family encloses the inner given circle (black). Twice the solution radius

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considering all possible combinations of internal and external tangency of the solution circle to the three given circles.

12629: 12340: 12053: 11456: 8250: 8004: 9273:

A Treatise on Conic Sections, Containing an Account of Some of the Most Important Modern Algebraic and Geometric Methods

1070:, Apollonius' problem has ten special cases, depending on the nature of the three given objects, which may be a circle ( 12542: 12120: 10947: 10532: 9375: 8045: 5796: 482:

Although successful in solving Apollonius' problem, van Roomen's method has a drawback. A prized property in classical

172: 12520: 7909:. The Apollonian gasket also has deep connections to other fields of mathematics; for example, it is the limit set of 7089:

line of any two. Hence, the center is the point of intersection of any two perpendicular bisectors. Similarly, in the

5926:, respectively. In the latter case, no point on either of the two lines has real coordinates except for the origin . 5553:{\displaystyle \{\in \mathbf {P} ^{2}\colon AX^{2}+BXY+CY^{2}+DXZ+EYZ+FZ^{2}=0\}\leftrightarrow \in \mathbf {P} ^{5}.} 735:. Van Roomen solved the general problem by solving a simpler problem, that of finding the circles that are tangent to 12998: 12332: 10837: 10251: 10195: 9351: 9280: 8997: 8785: 8723: 8692: 8636: 8585: 4490:

indicated in Figure 8. Thus, all eight solutions of the general Apollonius problem can be found by this method.

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case (a circle, a line and a point) using three lemmas. Again shrinking one circle to a point, Viète transformed the

5790:, these equations also demonstrate that every conic passing through the circular points has an equation of the form 12928: 12753: 6941:{\displaystyle \Psi =\{(D_{1},D_{2},D_{3},C)\in (\mathbf {P} ^{3})^{4}\colon C{\text{ is tangent to all }}D_{i}\}.} 4472:

This formula represents four solutions, corresponding to the two choices of the sign of θ, and the two choices for

10624:

Lewis RH, Bridgett S (2003). "Conic Tangency Equations and Apollonius Problems in Biochemistry and Pharmacology".

9387:

Milorad R. Stevanovic, Predrag B. Petrovic, and Marina M. Stevanovic, "Radii of circles in Apollonius' problem",

8745:(1600). "Apollonius Gallus. Seu, Exsuscitata Apolloni Pergæi Περι Επαφων Geometria". In Frans van Schooten (ed.). 7385:

The problem of counting the number of solutions to different types of Apollonius' problem belongs to the field of

6573:

is a quotient of quadratics, neither of which vanishes identically. Therefore, it vanishes at two points and has

4690:, respectively. Gergonne's solution aims to locate these six points, and thus solve for the two solution circles. 12948: 12850: 12748: 12569: 10441: 7414:, or when one or three of the given circles are themselves solutions. (An example of the latter is given in the 973:

to the three given points have known values. These four points correspond to the center of the solution circle (

12089: 12022: 11655: 11535: 10288:"Solutio facilis problematis, quo quaeritur sphaera, quae datas quatuor sphaeras utcunque dispositas contingat" 10149: 9755: 8823: 8234: 8156: 4634:

Gergonne's approach is to consider the solution circles in pairs. Let a pair of solution circles be denoted as

584: 140:) and a classification of solutions according to 33 essentially different configurations of the given circles. 10076: 12833: 10354:

Français J (January 1813). "Solution analytique du problème de la sphère tangente à quatre sphères données".

7962:

in 1936, several people solved (independently) the mutually tangent case corresponding to Soddy's circles in

4648:(the pink circles in Figure 6), and let their tangent points with the three given circles be denoted as 8040:

from the center of attraction and observations of tangent lines to the orbit corresponding to instantaneous

4024:

Figure 6: A conjugate pair of solutions to Apollonius's problem (pink circles), with given circles in black.

1800:{\displaystyle \left(x_{s}-x_{3}\right)^{2}+\left(y_{s}-y_{3}\right)^{2}=\left(r_{s}-s_{3}r_{3}\right)^{2}.} 128:

used this symmetry to provide an elegant straightedge and compass solution, while other mathematicians used

13008: 12743: 12158: 11915: 11803: 10978: 10888: 10327: 6172:{\displaystyle \Phi =\{(r,C)\in D\times \mathbf {P} ^{3}\colon C{\text{ is tangent to }}D{\text{ at }}r\}.} 1641:{\displaystyle \left(x_{s}-x_{2}\right)^{2}+\left(y_{s}-y_{2}\right)^{2}=\left(r_{s}-s_{2}r_{2}\right)^{2}} 1485:{\displaystyle \left(x_{s}-x_{1}\right)^{2}+\left(y_{s}-y_{1}\right)^{2}=\left(r_{s}-s_{1}r_{1}\right)^{2}} 8447: 4169:

When two of the given circles are concentric, Apollonius's problem can be solved easily using a method of

147:

have been studied. The configuration of three mutually tangent circles has received particular attention.

13003: 12993: 12938: 12797: 12594: 11867: 11798: 7107: 6679: 1209: 102: 12688: 9408: 643:

Figure 3: Two given circles (black) and a circle tangent to both (pink). The center-to-center distances

12738: 12733: 12656: 12559: 12525: 12488: 12347: 12234: 11494: 11310: 9795: 9791: 8891:

Methods and Theories for the Solution of Problems of Geometrical Constructions, Applied to 410 Problems

8655: 7995:. Similarly, the location of an aircraft may be determined from the difference in arrival times of its 1027: 476: 259:

Apollonius' problem can also be formulated as the problem of locating one or more points such that the

11285: 9166:(in French) (5th edition, revised and augmented ed.). Paris: Gauthier-Villars. pp. 252–256. 7422:.) However, there are no Apollonius problems with seven solutions. Alternative solutions based on the 5721:

Taking the sum and difference of these equations shows that it is equivalent to impose the conditions

263:

of its distances to three given points equal three known values. Consider a solution circle of radius

184:

be distinct, meaning that they do not coincide. Solutions to Apollonius' problem are sometimes called

12703: 12619: 12273: 12029: 12000: 11360: 11215: 8751:(in Latin). ex officina B. et A. Elzeviriorum (Lugduni Batavorum) (published 1646). pp. 325–346. 8064:

each of which touches several others. Finally, Apollonius' problem has been applied to some types of

6053:

is also equal to a circular point, this should be interpreted as the intersection multiplicity being

6033:, but some of these points might collide. Appolonius' problem is concerned with the situation where 6000: 5899: 4526: 129: 31:

Figure 2: Four complementary pairs of solutions to Apollonius's problem; the given circles are black.

10638: 8715: 8709: 8003:

problem is equivalent to the three-dimensional generalization of Apollonius' problem and applies to

7052:). As an example, the type of Apollonius problem with a given circle, line, and point is denoted as 2203:. That geometry represents circles, lines and points in a unified way, as a five-dimensional vector 2188: 1962: 1910: 494:. However, many such "impossible" problems can be solved by intersecting curves such as hyperbolas, 243:

The solution circle may be either internally or externally tangent to each of the given circles. An

12574: 12515: 12498: 12483: 12428: 12163: 11897: 11440: 10025: 9145: 8312: 7852: 7415: 5313: 4809: 4009: 3840: 3828: 248: 94: 9303:

Advanced Euclidean Geometry: An Elementary treatise on the geometry of the Triangle and the Circle

8215: 8044:. The special case of the problem of Apollonius when all three circles are tangent is used in the 4714:

were guaranteed to fall on it, those two points could be identified as the intersection points of

4598:

so that two of them are tangential (touching). Their point of tangency is chosen as the center of

4001: 1212:

of their centers. For example, the center positions of the three given circles may be written as (

12552: 12305: 12125: 12101: 11975: 11910: 11851: 11788: 11778: 11514: 11433: 11295: 11205: 11085: 7106:

tangent to three given circles. The nine other cases involving points and lines may be viewed as

4963: 4778: 4041: 4037: 1183:

case (two circles and a point), the latter case by two lemmas. Finally, Viète solved the general

121: 11395: 8449:

Exercices de géométrie, comprenant l'exposé des méthodes géométriques et 2000 questions résolues

7831:

Figure 13: A symmetrical Apollonian gasket, also called the Leibniz packing, after its inventor

4728:(Figure 6). The remaining four tangent points would be located similarly, by finding lines 4594:

In the second approach, the radii of the given circles are modified appropriately by an amount Δ

3539:{\displaystyle \left|\mathbf {c} _{1}-\mathbf {c} _{2}\right|^{2}=\left(r_{1}+r_{2}\right)^{2}.} 3408:{\displaystyle \left|\mathbf {c} _{1}-\mathbf {c} _{2}\right|^{2}=\left(r_{1}-r_{2}\right)^{2}.} 1847:. Since the three signs may be chosen independently, there are eight possible sets of equations 12923: 12913: 12678: 12641: 12614: 12508: 12148: 12096: 11995: 11823: 11773: 11758: 11753: 11524: 11325: 11260: 11250: 11200: 10633: 10287: 10187: 9343: 9336: 8952: 8461: 8453: 8098: 8049: 7996: 7988: 7975: 7086: 5680: 5635: 4476:. The remaining four solutions can be obtained by the same method, using the substitutions for 3832: 1129: 430: 23:

Figure 1: A solution (in purple) to Apollonius's problem. The given circles are shown in black.

10829: 10127: 10121: 8777: 8769: 8457: 7906: 5984: 151:

gave a formula relating the radii of the solution circles and the given circles, now known as

12970: 12877: 12763: 12758: 12460: 12435: 12227: 12196: 12048: 11892: 11833: 11710: 11633: 11383: 11290: 11140: 9042: 8069: 7821: 7468: 7419: 5787: 4619:

Figure 9: The two tangent lines of the two tangent points of a given circle intersect on the

4599: 4062: 3959:{\displaystyle {\overline {\mathbf {OP} }}\cdot {\overline {\mathbf {OP^{\prime }} }}=R^{2}.} 1303: 1058:.) Thus, the solutions to Apollonius' problem are the intersections of a line with a circle. 623: 189: 152: 10320: 10179: 9137: 9119:(2nd edition, revised and enlarged ed.). New York: Barnes and Noble. pp. 222–227. 8304: 5567:

in the complex projective plane is defined to be a conic that passes through the two points

4615: 116:

Later mathematicians introduced algebraic methods, which transform a geometric problem into

12884: 12604: 12361: 12173: 12113: 12077: 11920: 11743: 11690: 11660: 11650: 11559: 11422: 11315: 11230: 11185: 11165: 11010: 10995: 10773: 10546: 10503: 10394: 9969: 9843: 9076: 8985: 8864: 8839: 8478: 8372: 7925: 7472: 7440: 7386: 7077: 6470:

There are two possibilities for the number of points of intersections. One is that either

6377: 4833:. To understand this reciprocal relationship, consider the two tangent lines to the circle 4630:

are the poles of the lines connecting the blue tangent points in each given circle (black).

4170: 3792: 1832: 1108: 615: 564: 560: 556: 454: 417:

again cancels out. The re-formulation in terms of center-center distances is useful in the

125: 74: 8746: 8: 12693: 12673: 12589: 12579: 12440: 12153: 12072: 12060: 12041: 12005: 11925: 11843: 11828: 11818: 11768: 11763: 11705: 11574: 11466: 11330: 11320: 11220: 11190: 11130: 11105: 11030: 11020: 11005: 10923: 10755: 9138: 8818: 8765: 8742: 8305: 8037: 8016: 7879: 7429: 7423: 6320: 5753: 5727: 5309: 3836: 2200: 1113: 1089: 1055: 588: 522: 450: 446: 225: 137: 98: 48: 10550: 10507: 9973: 9847: 9080: 8376: 7635:{\displaystyle (k_{1}+k_{2}+k_{3}+k_{s})^{2}=2(k_{1}^{2}+k_{2}^{2}+k_{3}^{2}+k_{s}^{2})} 12908: 12624: 12537: 12265: 12209: 12168: 12108: 12036: 11902: 11877: 11695: 11670: 11638: 11476: 11235: 11180: 11145: 11040: 10873: 10822: 10793: 10690: 10664: 10593: 10570: 10469: 10418: 10099: 10042: 10000: 9957: 9899: 9807: 9668: 9625: 9617: 9591: 9561: 9526: 9491: 9392: 9253: 9117:

College Geometry: An Introduction to the Modern Geometry of the Triangle and the Circle

9092: 8549: 8528: 8388: 8276: 8184: 7921: 6733:

are the same up to a change of coordinates, so this completely determines the shape of

5620: 5305: 4538: 4522: 4045: 4030: 3824: 3820: 2101: 1306: 707: 576: 483: 461:, who identified the centers of the solution circles as the intersection points of two 458: 422: 233: 133: 117: 86: 70: 36: 10854: 10647: 7410:, an odd number of solutions is possible in some cases, e.g., the single solution for 3556:

that belong to the Lie quadric and are also orthogonal (perpendicular) to the vectors

2179:

The two roots of any quadratic equation may be of three possible types: two different

992: 19: 12867: 12839: 12721: 12698: 12646: 12564: 12450: 12204: 11882: 11793: 11643: 11569: 11542: 11305: 11125: 11115: 11050: 10970: 10870: 10833: 10781: 10707: 10694: 10604: 10562: 10461: 10422: 10382: 10247: 10191: 10180: 10155: 10131: 10117: 10046: 10005: 9987: 9891: 9811: 9683: 9679: 9660: 9609: 9530: 9495: 9371: 9347: 9331: 9306: 9276: 9257: 9167: 9149: 9120: 8993: 8953:"Solutio facilis problematis, quo quaeritur circulus, qui datos tres circulos tangat" 8893:. London: Sampson Low, Marston, Searle & Rivington. pp. 94–95 (Example 403). 8781: 8719: 8688: 8663: 8632: 8581: 8344: 8316: 8283: 8254: 8176: 7959: 7902: 7890: 7863: 7832: 5246:

of one of the four lines connecting the homothetic centers. Finding the same pole in

4975: 4578: 2276: 2184: 2037: 1272:). Similarly, the radii of the given circles and a solution circle can be written as 1034:, and from their two known distance ratios, Newton constructs a line passing through 639: 507: 229: 156: 10103: 9706: 9629: 9565: 9096: 8392: 7974:

The principal application of Apollonius' problem, as formulated by Isaac Newton, is

7476: 4954: 4020: 1159:

case (a circle and two lines) by shrinking the circle into a point, rendering it an

1051: 599: 148: 12827: 12803: 12779: 12609: 12355: 12084: 11955: 11872: 11628: 11616: 11564: 11300: 10745: 10680: 10643: 10554: 10511: 10494: 10453: 10410: 10266: 10224: 10091: 10064: 10034: 9995: 9977: 9883: 9851: 9834: 9799: 9652: 9601: 9553: 9518: 9483: 9447: 9245: 9133: 9084: 8928: 8520: 8428: 8380: 8246: 8168: 8103: 8093: 7933: 7731: 5904:

There are two types of singular circles. One is the union of the line at infinity

4626:(red line) of the two solution circles (pink). The three points of intersection on 4123: 3990: 1117: 918: 753:. He noted that the center of a circle tangent to both given circles must lie on a 571:

theorem, Gergonne's method exploits the conjugate relation between lines and their

568: 491: 144: 10574: 10314: 8482: 8008: 7676:

and radius of the solution circle, respectively, and similarly for the curvatures

7048:, to denote whether the given elements are a circle, line or point, respectively ( 4493:

Any initial two disjoint given circles can be rendered concentric as follows. The

4077: 1148:, whereas the radius of its externally tangent given circles must be changed by −Δ 1144:, the radius of its internally tangent given circles must be likewise changed by Δ 12815: 12584: 12478: 11725: 11715: 11609: 11335: 10907: 10600: 10414: 9825: 9759: 8886: 8842:(January 1811). "Solutions de plusieurs problêmes de géométrie et de mécanique". 8205: 8065: 8053: 8000: 7955: 7847:. For the sphere, the problem is to construct all the circles (the boundaries of 7844: 7726: 4855: 598:

Algebraic solutions to Apollonius' problem were pioneered in the 17th century by

580: 572: 398:; they depend only on the known radii of the given circles and not on the radius 368:, respectively. Therefore, differences in these distances are constants, such as 7936:, and many alternative solution methods have been developed over the centuries. 7377: 5780:

Therefore, the variety of all circles is a three-dimensional linear subspace of

12858: 12683: 12666: 12636: 12423: 11985: 11980: 11808: 11700: 11680: 11508: 11065: 11035: 10588: 10283: 10228: 10053:

Boyd, David W. (1973). "The Residual Set Dimension of the Apollonian Packing".

9579: 8948: 8367:

Schmidt, RO (1972). "A new approach to geometry of range difference location".

8238: 8012: 7910: 6574: 5317: 4230: 4040:—that intersects all of them perpendicularly; the center of that circle is the 4005: 3994: 3788: 3248: 2811: 2485: 1124:

case (a line and two points). Following Euclid a second time, Viète solved the

607: 10068: 9452: 9435: 9249: 8933: 8914: 8867:(1813–1814). "Recherche du cercle qui en touche trois autres sur une sphère". 5999:

intersect in four points total, when those points are counted with the proper

2510:

be two vectors belonging to this quadric; the norm of their difference equals

1082:). By custom, these ten cases are distinguished by three letter codes such as 12987: 12903: 12785: 12727: 11945: 11813: 11783: 11604: 11412: 11355: 10465: 10444:(1 January 1882). "The intersection of circles and intersection of spheres". 10366: 10342:

Français J (January 1810). "De la sphère tangente à quatre sphères données".

10310: 10175: 9991: 9917: 9895: 9775: 9664: 9643:

Oldknow A (1 April 1996). "The Euler–Gergonne–Soddy Triangle of a Triangle".

9613: 9436:"On the Number and nature of the Solutions of the Apollonian Contact Problem" 9236:

Knight RD (2005). "The Apollonius contact problem and Lie contact geometry".

9171: 9024: 9013: 8800: 8384: 8180: 8026:

Apollonius' problem has other applications. In Book 1, Proposition 21 in his

7886: 7885:

that is not known exactly but is roughly 1.3, which is higher than that of a

7856: 7848: 7722: 7452: 6208: 4462:{\displaystyle \theta =\pm 2\arctan \left({\sqrt {\frac {1-C}{1+C}}}\right).} 2815: 2295: 1015: 950: 619: 526: 503: 164: 27: 10939: 10785: 10711: 9803: 2183:, two identical real numbers (i.e., a degenerate double root), or a pair of 12791: 12661: 12470: 12445: 12408: 12250: 11887: 11675: 11350: 11110: 11100: 10750: 10685: 10668: 10566: 10533:"Geometry of locating sounds from differences in travel time: Isodiachrons" 10319:(in French). Paris: Imprimerie de Crapelet, chez J. B. M. Duprat. pp. 10145: 10009: 9982: 9871: 9207: 8624: 8604: 8573: 8057: 8033: 7797: 7060: 6669: 4979: 4813: 4620: 4494: 3056:= 1 for circles, the product of any two such vectors on the quadric equals 946: 917:. This property, of having a fixed difference between the distances to the 611: 466: 426: 168: 106: 10095: 10077:"Hausdorff dimension and conformal dynamics III: Computation of dimension" 6699:

in two points. Together, these two possibilities for the intersection of

4854:

with the solution circles; the intersection of these tangent lines is the

4586:

different given circles may be shrunk to a point for different solutions.

4505:

on this radical axis, two circles can be constructed that are centered on

3803: 12809: 12418: 12297: 11501: 11389: 11095: 11080: 10055: 10023:

Boyd, David W. (1973). "Improved Bounds for the Disk Packing Constants".

8432: 8134:

100 Great Problems of Elementary Mathematics: Their History and Solutions

8061: 7984: 7944: 3787:

The advantage of this re-statement is that one can exploit theorems from

2481: 2299: 2180: 79: 8338: 6652:. These are precisely the points at which the circle whose equation is 6269:

cut out by two linear equations in the space of circles. Consequently,

4974:

of the unknown solution circles as follows. Any pair of circles has two

1851:, each set corresponding to one of the eight types of solution circles. 1187:

case (three circles) by shrinking one circle to a point, rendering it a

514:

showed that the problem can be solved by using the intersections of two

12933: 12821: 11487: 11406: 11345: 11340: 11280: 11265: 11210: 11195: 11150: 11090: 11075: 11055: 11025: 10990: 10473: 10038: 9935:

Vannson (1855). "Contact des cercles sur la sphère, par la geométrie".

9903: 9672: 9621: 9557: 9522: 9487: 8532: 8188: 7363: 7340: 7317: 7294: 7271: 7248: 7225: 7202: 7179: 7156: 4497:

of the two given circles is constructed; choosing two arbitrary points

4058: 4008:. The planar Apollonius problem can be transferred to the sphere by an 592: 511: 217: 10558: 9856: 9829: 7430:

Mutually tangent given circles: Soddy's circles and Descartes' theorem

1042:

must lie. However, the ratio of distances TZ/TA is also known; hence,

11240: 11225: 11175: 11070: 11060: 11045: 11015: 10878: 10516: 10489: 9687: 9596: 9088: 8020: 7673: 7395: 4173:. The radii of the three given circles are known, as is the distance 3287:

0—then their corresponding circles are tangent. For if the two signs

1855: 1854:

The general system of three equations may be solved by the method of

754: 711: 462: 90: 10457: 10398: 10212: 9887: 9779: 9752: 9656: 9605: 9471: 9067:

Hoshen J (1996). "The GPS Equations and the Problem of Apollonius".

8631:. Cambridge: Cambridge University Press. pp. 162–165, 238–241. 8524: 8172: 7827: 908:

between these distances is always a constant that is independent of

490:. Many constructions are impossible using only these tools, such as 155:. Solving Apollonius' problem iteratively in this case leads to the 12455: 12396: 12219: 11377: 11155: 11000: 8081: 8041: 7905:

in the 17th century, and is a curved precursor of the 20th-century

7458: 6294:

must be tangent at only a single point. Therefore, the projection

4517:

renders the constructed circles into straight lines emanating from

1014:

Instead of solving for the two hyperbolas, Newton constructs their

515: 499: 8056:'s contour for complex integration, given by the boundaries of an 8036:

used his solution of Apollonius' problem to construct an orbit in

5069:. It follows, therefore, that the products of distances are equal 4607:

produces a solution circle tangent to the original three circles.

4061:. Under this inversion, the solution circles must fall within the 12413: 12401: 12289: 11950: 11275: 11270: 11170: 11160: 11135: 10731:"Apollonius' Problem: A Study of Solutions and Their Connections" 8511:

Bruen A, Fisher JC, Wilker JB (1983). "Apollonius by Inversion".

7875: 7024: 5296:, there is no pair of solutions for that homothetic-center line. 4872:. Since the distances from that pole point to the tangent points 3854:

consists of the following operation (Figure 5): every point

2484:

is defined as those vectors whose product with themselves (their

892:, again depending on their chosen tangency. Thus, the difference 495: 160: 44: 10868: 10154:(2nd ed.). New York: Taylor and Francis. pp. 131–138. 8685:

What is Mathematics? An Elementary Approach to Ideas and Methods

7447:. This special case of Apollonius' problem is also known as the 4036:

In general, any three distinct circles have a unique circle—the

1018:

instead. For any hyperbola, the ratio of distances from a point

12967: 11245: 11120: 10810:Über die Entwicklung der Elementargeometrie im XIX. Jahrhundert 9211: 9161: 8339:

Hofmann-Wellenhof B, Legat K, Wieser M, Lichtenegger H (2003).

8077: 7840: 7072: 6277:. Since it is possible to exhibit a circle that is tangent to 3831:; however, they do not simplify the problem, since they merely 2279:

of the circle, with counterclockwise circles having a positive

1103: 1047: 1046:

also lies on a known circle, since Apollonius had shown that a

9729:

Beecroft H (1842). "Properties of Circles in Mutual Contact".

9344:

48–51 (Apollonius' problem), 60 (extension to tangent spheres)

8687:. London: Oxford University Press. pp. 125–127, 161–162. 6043:, meaning that the intersection multiplicity at that point is 4156:

of the inner and outer radii, while twice its center distance

4110:

of the inner and outer radii, while twice its center distance

2199:

The same algebraic equations can be derived in the context of

935:, where the internal or external tangency of the solution and 11621: 11599: 11255: 10926:. Feature Column at the American Mathematical Society website 9275:. London: Longmans, Green and Co. pp. 110–115, 291–292. 8629:

The Mathematical Papers of Isaac Newton, Volume VI: 1684–1691

8578:

The Mathematical Papers of Isaac Newton, Volume VI: 1684–1691

7992: 7796:

Sundry extensions of Descartes' theorem have been derived by

7721:

Descartes' theorem was rediscovered independently in 1826 by

6182:

For a curve that is the vanishing locus of a single equation

4886:

are equal, this pole point must also lie on the radical axis

434: 214: 110: 9713:, (C. Adam and P. Tannery, Eds.), Paris: Leopold Cert 1901. 9544:

Fitz-Gerald JM (1974). "A Note on a Problem of Apollonius".

9020:(in French). Paris: Unknown publisher. pp. No. 158–159. 8076:

and the design of pharmaceuticals that bind in a particular

7451:. The three given circles of this Apollonius problem form a 6461:

are precisely the circles in the pencil that are tangent to

2049:

into the linear formulae yields the corresponding values of

3265:

are orthogonal (perpendicular) to one another—that is, if (

10824:

The Penguin Dictionary of Curious and Interesting Geometry

9582:(1 January 2001). "Tangent Spheres and Triangle Centers". 9031:(in French). Paris: Unknown publisher. pp. 390, §334. 7839:

Apollonius' problem can be extended from the plane to the

5941:

is any other circle, then, by the definition of a circle,

3247:

represent the length of that difference vector, i.e., the

9045:(July 1806). "Du cercle tangent à trois cercles donnés". 8960:

Nova Acta Academiae Scientiarum Imperialis Petropolitanae

8660:

A sequel to the first six books of the Elements of Euclid

8073: 7729:. Soddy published his findings in the scientific journal 4589: 457:. The first new solution method was published in 1596 by 55:

190 BC) posed and solved this famous problem in his work

10276: 9368:

3264 and All That: A Second Course in Algebraic Geometry

4552: 4541:

can be increased significantly by resizing. As noted in

1254:), whereas that of a solution circle can be written as ( 449:

has been lost, but reconstructions have been offered by

10595:

Modular functions and Dirichlet series in number theory

8568: 8566: 8419:

Althiller-Court N (1961). "The problem of Apollonius".

8414: 8412: 8410: 8408: 8406: 8404: 8402: 8132:

Dörrie H (1965). "The Tangency Problem of Apollonius".

6682:, there are precisely two branch points, and therefore 6521:, then so is every circle in the pencil, and therefore 4561:. In that case, Apollonius' problem degenerates to the 583:

in 1879; one example is the annular solution method of

10669:"The osculatory packing of a three-dimensional sphere" 10295:

Mémoires de l'Académie des Sciences de St.-Pétersbourg

8580:. Cambridge: Cambridge University Press. p. 164. 4958:

Figure 10: The poles (red points) of the radical axis

693:, respectively, so their difference is independent of 10246:. London: Macmillan. pp. §383–396, pp. 244–251. 9069:

IEEE Transactions on Aerospace and Electronic Systems

8915:"Configurations of Cycles and the Apollonius Problem" 8828:. Oxford: Clarendon Press. pp. 181–185, 416–417. 8369:

IEEE Transactions on Aerospace and Electronic Systems

7488: 6825: 6095: 5799: 5756: 5730: 5683: 5638: 5357: 5078: 4401: 4242: 3899: 3586: 3445: 3314: 3065: 2827: 2814:

over addition and subtraction (more precisely, it is

2519: 2311: 1965: 1913: 1657: 1501: 1345: 996:

The set of points with a constant ratio of distances

453:

and others, based on the clues in the description by

10772: 10706:(in French). Paris: Albin Michel. pp. 219–226. 10701: 9393:

http://forumgeom.fau.edu/FG2017volume17/FG201735.pdf

9360: 8776:(2nd ed.). John Wiley & Sons, Inc. p. 8563: 8399: 7407: 1026:

and to the directrix is a fixed constant called the

10174: 8814:(in German). Copenhagen: Unknown. pp. 381–383. 8794: 8334: 8332: 8233: 7901:= 2). The Apollonian gasket was first described by 7725:, in 1842 by Philip Beecroft, and again in 1936 by 7426:have been developed and used in higher dimensions. 3874:are collinear, and the product of the distances of 47:to three given circles in a plane (Figure 1). 10821: 10592: 10371:Correspondance sur l'École Impériale Polytechnique 10356:Correspondance sur l'École Impériale Polytechnique 10344:Correspondance sur l'École Impériale Polytechnique 10186:. Cambridge: Cambridge University Press. pp. 9402: 9400: 9335: 9114: 8844:Correspondance sur l'École Impériale Polytechnique 8544: 8542: 8439: 8341:Navigation: Principles of Positioning and Guidance 8275: 8229: 8227: 8225: 7634: 6940: 6171: 6003:. That is, there are four points of intersection 5881: 5768: 5742: 5710: 5668: 5552: 5342:parametrized by five-dimensional projective space 5206: 4461: 4377: 4015: 3958: 3776: 3538: 3407: 3220: 3003: 2799: 2469: 2000: 1948: 1799: 1640: 1484: 12944:Statal Institute of Higher Education Isaac Newton 9440:Proceedings of the Edinburgh Mathematical Society 9132: 8711:Famous problems of geometry and how to solve them 8556:(in Latin). Würzburg: Typis Georgii Fleischmanni. 8510: 8418: 5882:{\displaystyle (X-aZ)^{2}+(Y-bZ)^{2}=r^{2}Z^{2},} 3251:. This formula shows that if two quadric vectors 3042:) = 0 (both belong to the Lie quadric) and since 12985: 10524: 10399:"De la sphère tangente à quatre sphères donnèes" 9724: 9722: 9110: 9108: 9106: 8859: 8857: 8329: 2291:is zero for a straight line, and one otherwise. 2271:may be positive or negative; for visualization, 1858:. When multiplied out, all three equations have 73:, but a 4th-century AD report of his results by 10778:Pappus d'Alexandrie: La collection mathématique 10530: 10428: 10403:Journal für die reine und angewandte Mathematik 9924:. Paris: Unknown publisher. pp. 415, §356. 9784:Journal für die reine und angewandte Mathematik 9770: 9768: 9537: 9397: 9296: 9294: 9292: 9018:De la corrélation dans les figures de géométrie 8863: 8539: 8445: 8222: 8159:(1 January 1968). "The Problem of Apollonius". 7978:, which seeks to determine a position from the 4542: 3819:A natural setting for problem of Apollonius is 845:between the centers of the solution circle and 791:between the centers of the solution circle and 16:Geometry problem about finding touching circles 10812:(in German). Berlin: Teubner. pp. 97–105. 10244:An elementary treatise on modern pure geometry 10217:Proceedings of the London Mathematical Society 9543: 9406: 9326: 9324: 9322: 9231: 9229: 9062: 9060: 8908: 8906: 8904: 8902: 8900: 8758: 8682: 8599: 8597: 8362: 8360: 8311:. New York: Oxford University Press. pp. 8151: 8149: 8147: 8145: 8143: 8068:, which arise in disparate fields such as the 7113: 7025:Ten combinations of points, circles, and lines 6428:correspond to the circles whose equations are 5893:lines. The non-degenerate circles are called 5017:be a center of similitude for the two circles 977:) and the centers of the three given circles ( 969:, such that the differences in distances from 486:is the ability to solve problems using only a 163:to be described in print, and is important in 120:. These methods were simplified by exploiting 12235: 10969: 10955: 10921: 10804:Trans., introd., and notes by Paul Ver Eecke. 10623: 9955: 9719: 9465: 9463: 9427: 9202: 9200: 9103: 8980: 8978: 8879: 8854: 8764: 8737: 8735: 8650: 8648: 8506: 8504: 8502: 8500: 8498: 8496: 8494: 8484:Pappi Alexandrini collectionis quae supersunt 8127: 8125: 8123: 8121: 8119: 7049: 6283:at only a single point, a generic element of 5235:is defined by two points: the radical center 784:, respectively (Figure 3). The distance 465:. Van Roomen's method was refined in 1687 by 10728: 10538:Journal of the Acoustical Society of America 10481: 10326: 10235: 10116: 9910: 9765: 9509:Study E (1897). "Das Apollonische Problem". 9289: 9178: 9006: 8838: 8812:Die Lehre von den Kegelschnitten im Altertum 8548: 8273: 7939:Apollonius' problem can even be extended to 7399: 6932: 6832: 6672:of this morphism are the circles tangent to 6163: 6102: 5487: 5358: 4770:, respectively. To construct a line such as 4546: 4208:are known (Figure 7), and the distance 4000:Circle inversions correspond to a subset of 1102:case (three points) following the method of 418: 10718: 10369:(January 1813). "Mémoire sur les sphères". 10204: 9928: 9874:(1 June 1967). "On a theorem in geometry". 9433: 9319: 9264: 9226: 9057: 8912: 8897: 8826:, Volume II: From Aristarchus to Diophantus 8610:Philosophiæ Naturalis Principia Mathematica 8594: 8366: 8357: 8204: 8155: 8140: 7954:hyperspheres. Following the publication of 7136:(solution in pink; given objects in black) 7002:, the generic fiber of the projection from 6338:, is also irreducible and two dimensional. 6083:of all circles. To see this, consider the 5239:of the three given circles and the pole in 3807:Figure 5: Inversion in a circle. The point 2294:In this five-dimensional world, there is a 1030:. The two directrices intersect at a point 710:(1596) is based on the intersection of two 563:(1814). Whereas Poncelet's proof relies on 538: 531: 178: 63: 56: 12242: 12228: 10962: 10948: 10798:: CS1 maint: location missing publisher ( 10738:American Journal of Undergraduate Research 10587: 10353: 10341: 10143: 9864: 9818: 9742: 9728: 9578: 9502: 9460: 9300: 9197: 9184: 9162:Rouché, Eugène; Ch de Comberousse (1883). 8975: 8885: 8809: 8732: 8645: 8491: 8302: 8200: 8198: 8136:. New York: Dover. pp. 154–160 (§32). 8116: 7115:Table 1: Ten Types of Apollonius' Problem 4052: 634: 10749: 10684: 10637: 10515: 10487: 10393: 10241: 10182:Indra's Pearls: The Vision of Felix Klein 10016: 9999: 9981: 9855: 9774: 9678: 9642: 9595: 9451: 9417:Acta Mathematica Universitatis Comenianae 9235: 9041: 8932: 8473: 8471: 7916:The configuration of a circle tangent to 6538:. The other possibility is that neither 6241:linear conditions on the coefficients of 6077:. This variety is a quadric cone in the 4784:lies on all three lines (Figure 6). 4532: 2040:. Substitution of the numerical value of 1061: 10905: 10816: 10807: 10663: 10452:(1): 25–44, with four pages of Figures. 10440: 10332:Correspondance sur l'École Polytechnique 10309: 10210: 10074: 9916: 9270: 9066: 9047:Correspondance sur l'École Polytechnique 9023: 9012: 8984: 8741: 8662:. Hodges, Figgis & co. p. 122. 8623: 8603: 8572: 8131: 7999:signal at four receiving stations. This 7826: 7789:The sum of the squares of all four bends 7457: 7376: 6577:at two points. These are the points in 4953: 4614: 4122: 4076: 4071: 4019: 3993:, "infinity" is defined in terms of the 3802: 2283:and clockwise circles having a negative 1088: 991: 638: 228:lines can be considered as tangent at a 26: 18: 10626:Mathematics and Computers in Simulation 10365: 10282: 9934: 9870: 9824: 9508: 9469: 9413:-space in view of enumerative geometry" 9206: 8947: 8654: 8211:A Treatise on the Circle and the Sphere 8195: 6774:be three circles. If the intersection 5897:, while the degenerate ones are called 1179:case (a circle and two points) and the 213:. (The word "tangent" derives from the 12986: 11991:Latin translations of the 12th century 10380: 9711:Œuvres de Descartes, Correspondance IV 9330: 9144:. New York: Springer Verlag. pp. 8707: 8477: 8468: 7785:Since zero bend's a dead straight line 7783:There's now no need for rule of thumb. 7781:Though their intrigue left Euclid dumb 7372: 7100:incircle and excircles of the triangle 6727:is a quadric cone. All such cones in 5299: 4787:To locate a second point on the lines 4693:Gergonne's insight was that if a line 4610: 4590:Resizing two given circles to tangency 2194: 1194: 714:. Let the given circles be denoted as 492:dividing an angle in three equal parts 237: 93:, but this solution does not use only 89:solved the problem using intersecting 12494:Newton's law of universal gravitation 12223: 11721:Straightedge and compass construction 10943: 10869: 9956:Kasner E, Supnick F (December 1943). 8920:Rocky Mountain Journal of Mathematics 6189:, the condition that the curve meets 6071:be the variety of circles tangent to 4553:Shrinking one given circle to a point 205:; the intersection point is called a 188:, although the term is also used for 12652:Newton's theorem of revolving orbits 12249: 11686:Incircle and excircles of a triangle 10924:"When kissing involves trigonometry" 10828:. New York: Penguin Books. pp. 10390:(in German). Leipzig: B. G. Teubner. 10052: 10022: 9743:Beecroft H (1846). "Unknown title". 9446:: 135–147, attached figures 44–114. 9370:. Cambridge University Press, 2016. 8214:. Oxford: Clarendon Press. pp. 7897:= 1) but less than that of a plane ( 7762:Each gets three kisses from without. 7455:tangent to the two Soddy's circles. 7031:Special cases of Apollonius' problem 5623:of all circles is the subvariety of 4944:, giving the needed second point on 4581:of the two circles. Re-inversion in 4572:. Inversion in a circle centered on 3798: 3577:corresponding to the given circles. 3231:where the vertical bars sandwiching 1120:theorem, which he used to solve the 101:found such a solution by exploiting 69:, "Tangencies"); this work has been 12600:Leibniz–Newton calculus controversy 12341:standing on the shoulders of giants 10307:, series 1, volume 26, pp. 334–343. 10126:. New York: W. H. Freeman. p. 9958:"The Apollonian Packing of Circles" 9780:"Einige geometrische Betrachtungen" 8972:, series 1, volume 26, pp. 270–275. 8005:global navigation satellite systems 7356:three circles (the classic problem) 6951:There is a morphism which projects 6422:, respectively, then the points on 2259:) is the center of the circle, and 629: 13: 10657: 9938:Nouvelles Annales de Mathématiques 8748:Francisci Vietae Opera mathematica 7878:, being self-similar and having a 7803: 7787:And concave bends have minus sign, 7756:To bring this off the four must be 7752:'Tis not so when four circles kiss 6826: 6096: 5060:are pairs of antihomologous points 4547:method for inverting to an annulus 4350: 4347: 4344: 4332: 4310: 4290: 4287: 4284: 4264: 3977:lies within, and vice versa. When 3929: 3846:Inversion in a circle with center 3744: 3741: 3738: 3700: 3697: 3694: 3656: 3653: 3650: 3625: 3622: 3619: 3604: 3601: 3598: 1067: 14: 13020: 10847: 10384:Synthetische Geometrie der Kugeln 9645:The American Mathematical Monthly 9584:The American Mathematical Monthly 9366:Eisenbud, David and Harris, Joe, 8817: 8282:(2nd ed.). New York: Wiley. 8161:The American Mathematical Monthly 8015:varies with direction (i.e., the 7791:Is half the square of their sum. 7773:Four circles to the kissing come. 7764:If three in one, then is that one 7264:one circle, one line, and a point 6392:of circles. If the equations of 6376:. These two circles determine a 5953:intersect at the circular points 4187:, the angle θ, and the distances 2263:is its (non-negative) radius. If 272:and three given circles of radii 43:is to construct circles that are 12929:Isaac Newton Group of Telescopes 12203: 12190: 10704:Célèbres problèmes mathématiques 10213:"Contacts of Systems of Circles" 9746:The Lady's and Gentleman's Diary 9732:The Lady's and Gentleman's Diary 9409:"Apollonius' contact problem in 8487:(in Latin) (3 volumes ed.). 7758:As three in one or one in three. 7362: 7339: 7316: 7293: 7270: 7247: 7224: 7201: 7178: 7155: 7019: 6891: 6323:. It follows that the image of 6131: 5537: 5387: 4970:Gergonne found the radical axis 4010:inverse stereographic projection 3985:, the inversion is said to send 3925: 3921: 3906: 3903: 3469: 3454: 3338: 3323: 3131: 3116: 2716: 2701: 2676: 2661: 2411: 2396: 1010:to two fixed points is a circle. 408:that the solution-circle radius 12949:Newton International Fellowship 12630:generalized Gauss–Newton method 12543:Newton's method in optimization 10673:Canadian Journal of Mathematics 10617: 10581: 10446:American Journal of Mathematics 10260: 10168: 10110: 10084:American Journal of Mathematics 9949: 9700: 9636: 9572: 9391:17 (2017), 359–372: Theorem 1. 9381: 9035: 8941: 8832: 8714:. Dover Publications. pp. 8701: 8676: 8617: 8613:. Book I, Section IV, Lemma 16. 7969: 7777:The bend is just the inverse of 7760:If one in three, beyond a doubt 7748:For pairs of lips to kiss maybe 7471:. In a 1643 letter to Princess 7424:geometry of circles and spheres 6613:of degree two. The fiber over 5062:, and their lines intersect at 4700:could be constructed such that 4016:Pairs of solutions by inversion 159:, which is one of the earliest 12023:A History of Greek Mathematics 11536:The Quadrature of the Parabola 10151:The Pursuit of Perfect Packing 10123:The Fractal Geometry of Nature 8824:A History of Greek Mathematics 8296: 8267: 8046:Hardy–Littlewood circle method 7629: 7557: 7542: 7489: 7071:), which were solved first by 6902: 6886: 6880: 6835: 6812:is finite, then it has degree 6747:To conclude the argument, let 6556:. In this case, the function 6117: 6105: 5935:be a fixed smooth circle. If 5844: 5828: 5816: 5800: 5529: 5493: 5490: 5379: 5361: 5348:, where the correspondence is 5283:does not intersect its circle 3973:lies outside the circle, then 2001:{\displaystyle y_{s}=P+Qr_{s}} 1949:{\displaystyle x_{s}=M+Nr_{s}} 173:Hardy–Littlewood circle method 1: 10724:(in Latin). Gothae: Ettinger. 10702:Callandreau, Édouard (1949). 10648:10.1016/S0378-4754(02)00122-2 10178:, Series C, Wright D (2002). 9407:Dreschler K, Sterz U (1999). 8683:Courant R, Robbins H (1943). 8110: 7943:dimensions, to construct the 7924:, itself a generalization of 7779:The distance from the center. 7067:) or tangent to three lines ( 6919: is tangent to all 6370:, not necessarily tangent to 6247:. This shows that, for each 5228:In summary, the desired line 4033:, by the following argument. 3418:Conversely, if the two signs 2036:, which can be solved by the 1128:case (three lines) using the 604:Princess Elisabeth of Bohemia 565:homothetic centers of circles 12570:Newton's theorem about ovals 11804:Intersecting secants theorem 10922:Austin, David (March 2006). 10729:Gisch D, Ribando JM (2004). 10415:10.1515/crelle-1848-18483704 10075:McMullen, Curtis T. (1998). 9472:"Zum Problem des Apollonius" 9212:"The missing seventh circle" 8913:Zlobec BJ, Kosta NM (2001). 8627:(1974). DT Whiteside (ed.). 8576:(1974). DT Whiteside (ed.). 8009:GPS#Geometric interpretation 7102:formed by the three lines). 6273:is irreducible of dimension 5323:Every quadratic equation in 5199: 5167: 5135: 5103: 4926:. Thus, if we can construct 4840:drawn at its tangent points 4812:between those lines and the 3935: 3910: 2131:) is a solution, with signs 192:associated with Apollonius. 7: 12939:Sir Isaac Newton Sixth Form 12595:Corpuscular theory of light 12521:Schrödinger–Newton equation 11799:Intersecting chords theorem 11666:Doctrine of proportionality 9140:Geometry: Euclid and Beyond 9115:Altshiller-Court N (1952). 8087: 7775:The smaller are the benter. 6352:, fix two distinct circles 3858:is mapped into a new point 3815:with respect to the circle. 1878:on the left-hand side, and 1098:Viète began by solving the 575:in a circle. Methods using 539: 130:geometrical transformations 64: 10: 13025: 12348:Notes on the Jewish Temple 11495:On the Sphere and Cylinder 11448:On the Sizes and Distances 10599:(2nd ed.). New York: 9962:Proc. Natl. Acad. Sci. USA 7947:tangent to a given set of 7766:Thrice kissed internally. 7028: 6341:To determine the shape of 3829:Euclidean plane isometries 3811:' is the inverse of point 1112:. From this, he derived a 488:compass and a straightedge 440: 12957: 12894: 12849: 12772: 12714: 12469: 12389: 12324: 12257: 12197:Ancient Greece portal 12186: 12136: 12014: 12001:Philosophy of mathematics 11971: 11964: 11938: 11916:Ptolemy's table of chords 11860: 11842: 11741: 11734: 11590: 11552: 11369: 10977: 10971:Ancient Greek mathematics 10069:10.1112/S0025579300004745 9453:10.1017/S0013091500031898 9250:10.1007/s00022-005-0009-x 8481:(1876). F Hultsch (ed.). 7754:Each one the other three. 7750:Involves no trigonometry. 7218:one circle and two points 7094:solutions to the general 6955:onto its final factor of 6149: is tangent to 6001:intersection multiplicity 5711:{\displaystyle A-Bi-C=0.} 5669:{\displaystyle A+Bi-C=0,} 5593:denotes a square root of 5304:The techniques of modern 4819:of the solution circles, 4527:bipolar coordinate system 3791:on the maximum number of 1175:case. He then solved the 1163:case. He then solved the 591:, which was developed by 532: 57: 12999:Euclidean plane geometry 12499:post-Newtonian expansion 12379:Corruptions of Scripture 12371:Ancient Kingdoms Amended 11868:Aristarchus's inequality 11441:On Conoids and Spheroids 10531:Spiesberger, JL (2004). 10431:The Mathematical Monthly 10229:10.1112/plms/s1-23.1.135 10026:Aequationes Mathematicae 9753:MathWords online article 9187:The Mathematical Monthly 8774:A History of Mathematics 8446:Gabriel-Marie F (1912). 8385:10.1109/TAES.1972.309614 8278:Introduction to Geometry 7976:hyperbolic trilateration 7958:'s re-derivation of the 7853:stereographic projection 7402:to Apollonius' problem. 7394:solution circles form a 7310:one circle and two lines 7010: 6085:incidence correspondence 5314:complex projective plane 4996:and the line defined by 4165:equals their difference. 2167:), with opposite signs − 957:from three given points 587:. Another approach uses 179:Statement of the problem 95:straightedge and compass 37:Euclidean plane geometry 12689:Absolute space and time 12553:truncated Newton method 12526:Newton's laws of motion 12489:Newton's law of cooling 11976:Ancient Greek astronomy 11789:Inscribed angle theorem 11779:Greek geometric algebra 11434:Measurement of a Circle 10855:"Ask Dr. Math solution" 10271:Varia opera mathematica 9804:10.1515/crll.1826.1.161 9216:Elemente der Mathematik 8934:10.1216/rmjm/1020171586 8805:Mathematical Collection 8421:The Mathematics Teacher 8307:Visual Complex Analysis 7287:two circles and a point 7172:one line and two points 6680:Riemann–Hurwitz formula 6438:, where is a point of 5786:. After rescaling and 5316:. Solutions involving 4930:, we can find its pole 4919:must conversely lie on 4810:reciprocal relationship 4053:Inversion to an annulus 2298:product similar to the 739:given circles, such as 635:Intersecting hyperbolas 544:, "Tangencies"; Latin: 12924:Isaac Newton Telescope 12914:Isaac Newton Institute 12684:Newton–Puiseux theorem 12679:Parallelogram of force 12667:kissing number problem 12657:Newton–Euler equations 12560:Gauss–Newton algorithm 12509:gravitational constant 12210:Mathematics portal 11996:Non-Euclidean geometry 11951:Mouseion of Alexandria 11824:Tangent-secant theorem 11774:Geometric mean theorem 11759:Exterior angle theorem 11754:Angle bisector theorem 11458:On Sizes and Distances 10751:10.33697/ajur.2004.010 10686:10.4153/CJM-1973-030-5 9983:10.1073/pnas.29.11.378 9338:Excursions in Geometry 8869:Ann. Math. Pures Appl. 8768:, Merzbach UC (1991). 8454:Maison A. Mame et Fils 8371:. AES-8 (6): 821–835. 8070:error-correcting codes 8050:analytic number theory 7989:Decca Navigator System 7836: 7636: 7463: 7382: 7333:two circles and a line 7087:perpendicular bisector 6978:. This has dimension 6942: 6488:, is the equation for 6173: 5883: 5770: 5744: 5712: 5670: 5554: 5208: 4967: 4721:with the given circle 4631: 4543:Viète's reconstruction 4533:Resizing and inversion 4463: 4379: 4166: 4120: 4090:equals the difference 4025: 4002:Möbius transformations 3960: 3843:the original problem. 3816: 3778: 3540: 3409: 3222: 3005: 2801: 2471: 2267:is not zero, the sign 2002: 1950: 1801: 1642: 1486: 1095: 1062:Viète's reconstruction 1011: 703: 431:hyperbolic positioning 190:other types of circles 134:reflection in a circle 32: 24: 12878:Isaac Newton Gargoyle 12788: (nephew-in-law) 12764:Copernican Revolution 12759:Scientific Revolution 12620:Newton–Cotes formulas 12484:Newton's inequalities 12461:Structural coloration 11898:Pappus's area theorem 11834:Theorem of the gnomon 11711:Quadratrix of Hippias 11634:Circles of Apollonius 11582:Problem of Apollonius 11560:Constructible numbers 11384:Archimedes Palimpsest 10889:"Apollonius' Problem" 10874:"Apollonius' problem" 10316:Géométrie de position 10303:Reprinted in Euler's 10273:, p. 74, Tolos, 1679. 10096:10.1353/ajm.1998.0031 9922:Géométrie de position 9511:Mathematische Annalen 9476:Mathematische Annalen 9029:Géométrie de position 8968:Reprinted in Euler's 8807:, volume VII, p. 117. 8770:"Apollonius of Perga" 8274:Coxeter, HSM (1969). 7830: 7637: 7461: 7441:circumscribed circles 7380: 7195:two lines and a point 6943: 6620:is the set of points 6174: 5884: 5788:completing the square 5771: 5745: 5713: 5671: 5555: 5209: 4976:centers of similarity 4957: 4618: 4600:inversion in a circle 4464: 4388:Here, a new constant 4380: 4126: 4080: 4023: 3961: 3806: 3779: 3541: 3410: 3223: 3006: 2802: 2472: 2003: 1951: 1802: 1643: 1487: 1116:corresponding to the 1092: 995: 642: 624:Augustin Louis Cauchy 85:In the 16th century, 30: 22: 12885:Astronomers Monument 12575:Newton–Pepys problem 12548:Apollonius's problem 12516:Newton–Cartan theory 12429:Newton–Okounkov body 12362:hypotheses non fingo 12351: (c. 1680) 12114:prehistoric counting 11911:Ptolemy's inequality 11852:Apollonius's theorem 11691:Method of exhaustion 11661:Diophantine equation 11651:Circumscribed circle 11468:On the Moving Sphere 10780:(in French). Paris. 10774:Pappus of Alexandria 10719:Camerer, JG (1795). 9684:"Four Coins Problem" 9434:Muirhead RF (1896). 8819:Heath, Thomas Little 8513:Mathematics Magazine 8452:(in French). Tours: 8433:10.5951/MT.54.6.0444 7486: 7473:Elizabeth of Bohemia 7387:enumerative geometry 6823: 6550:is the equation for 6444:. The points where 6093: 5797: 5754: 5728: 5681: 5636: 5355: 5308:, and in particular 5076: 4399: 4240: 3897: 3793:linearly independent 3584: 3443: 3312: 3063: 2825: 2517: 2309: 1963: 1911: 1839:= 1) or externally ( 1655: 1499: 1343: 953:: to locate a point 616:Carl Friedrich Gauss 561:Joseph Diaz Gergonne 557:Jean-Victor Poncelet 521:Van Roomen's friend 455:Pappus of Alexandria 126:Joseph Diaz Gergonne 75:Pappus of Alexandria 41:Apollonius's problem 13009:History of geometry 12694:Luminiferous aether 12642:Newton's identities 12615:Newton's cannonball 12590:Classical mechanics 12580:Newtonian potential 12441:Newtonian telescope 12200: • 12006:Neusis construction 11926:Spiral of Theodorus 11819:Pythagorean theorem 11764:Euclidean algorithm 11706:Lune of Hippocrates 11575:Squaring the circle 11331:Theon of Alexandria 11006:Aristaeus the Elder 10551:2004ASAJ..116.3168S 10508:1937Natur.139Q..62. 9974:1943PNAS...29..378K 9876:Amer. Math. Monthly 9848:1936Natur.137.1021S 9546:Journal of Geometry 9389:Forum Geometricorum 9301:Johnson RA (1960). 9238:Journal of Geometry 9164:Traité de géométrie 9081:1996ITAES..32.1116H 8810:Zeuthen HG (1886). 8377:1972ITAES...8..821S 8303:Needham, T (2007). 8099:Apollonius' theorem 8038:celestial mechanics 8017:transmission medium 7907:Sierpiński triangle 7874:. This gasket is a 7628: 7610: 7592: 7574: 7373:Number of solutions 7129:Number of solutions 7116: 7081:. For example, the 6998:also has dimension 6321:birational morphism 5769:{\displaystyle B=0} 5743:{\displaystyle A=C} 5310:intersection theory 5300:Intersection theory 5217:which implies that 4808:, Gergonne noted a 4611:Gergonne's solution 4320: 4300: 4274: 2201:Lie sphere geometry 2195:Lie sphere geometry 1307:quadratic equations 1195:Algebraic solutions 1056:bipolar coordinates 589:Lie sphere geometry 447:Apollonius of Perga 138:Lie sphere geometry 118:algebraic equations 49:Apollonius of Perga 13004:Incidence geometry 12994:Conformal geometry 12919:Isaac Newton Medal 12724: (birthplace) 12538:Newtonian dynamics 12436:Newton's reflector 11893:Menelaus's theorem 11883:Irrational numbers 11696:Parallel postulate 11671:Euclidean geometry 11639:Apollonian circles 11181:Isidore of Miletus 10871:Weisstein, Eric W. 10490:"The Kiss Precise" 10488:Gossett T (1937). 10242:Lachlan R (1893). 10039:10.1007/BF01838194 9830:"The Kiss Precise" 9758:2008-01-18 at the 9558:10.1007/BF01954533 9523:10.1007/BF01444366 9488:10.1007/BF01443201 9342:. Dover. pp. 8243:Geometry Revisited 7872:Apollonian packing 7866:, also known as a 7845:quadratic surfaces 7837: 7632: 7614: 7596: 7578: 7560: 7469:Descartes' theorem 7464: 7449:four coins problem 7420:Descartes' theorem 7383: 7114: 6961:. The fiber over 6938: 6500:is a line through 6235:; it is therefore 6223:vanishes to order 6201:with multiplicity 6169: 5879: 5766: 5740: 5708: 5666: 5621:projective variety 5550: 5306:algebraic geometry 5204: 4968: 4951:(Figure 10). 4632: 4579:homothetic centers 4537:The usefulness of 4523:Apollonian circles 4459: 4375: 4304: 4278: 4258: 4167: 4121: 4072:algebraic solution 4026: 3956: 3821:inversive geometry 3817: 3774: 3536: 3405: 3218: 3001: 2797: 2467: 1998: 1946: 1810:The three numbers 1797: 1638: 1482: 1096: 1012: 708:Adriaan van Roomen 704: 579:were pioneered by 484:Euclidean geometry 459:Adriaan van Roomen 423:Adriaan van Roomen 234:inversive geometry 218:present participle 186:Apollonius circles 153:Descartes' theorem 87:Adriaan van Roomen 33: 25: 12981: 12980: 12873: (sculpture) 12840:Abraham de Moivre 12794: (professor) 12722:Woolsthorpe Manor 12674:Newton's quotient 12647:Newton polynomial 12605:Newton's notation 12336: (1661–1665) 12217: 12216: 12182: 12181: 11934: 11933: 11921:Ptolemy's theorem 11794:Intercept theorem 11644:Apollonian gasket 11570:Doubling the cube 11543:The Sand Reckoner 10908:"Tangent Circles" 10808:Simon, M (1906). 10610:978-0-387-97127-8 10559:10.1121/1.1804625 10211:Larmor A (1891). 10161:978-1-4200-6817-7 10137:978-0-7167-1186-5 9857:10.1038/1371021a0 9312:978-0-486-46237-0 9271:Salmon G (1879). 9155:978-0-387-98650-0 9134:Hartshorne, Robin 9126:978-0-486-45805-2 8669:978-1-4181-6609-0 8350:978-3-211-00828-7 8322:978-0-19-853446-4 8289:978-0-471-50458-0 8260:978-0-88385-619-2 7960:Descartes theorem 7926:Ptolemy's theorem 7903:Gottfried Leibniz 7864:Apollonian gasket 7833:Gottfried Leibniz 7443:, and are called 7370: 7369: 6920: 6716:demonstrate that 6494:. In this case, 6380:, meaning a line 6158: 6150: 5202: 5170: 5138: 5106: 4450: 4449: 4357: 4119:equals their sum. 3989:to infinity. (In 3938: 3913: 3886:equal the radius 3799:Inversive methods 2185:complex conjugate 2038:quadratic formula 508:doubling the cube 230:point at infinity 211:point of tangency 157:Apollonian gasket 13016: 12969: 12864: (monotype) 12828:William Stukeley 12824: (disciple) 12804:Benjamin Pulleyn 12780:Catherine Barton 12699:Newtonian series 12610:Rotating spheres 12356:General Scholium 12251:Sir Isaac Newton 12244: 12237: 12230: 12221: 12220: 12208: 12207: 12195: 12194: 12193: 11969: 11968: 11956:Platonic Academy 11903:Problem II.8 of 11873:Crossbar theorem 11829:Thales's theorem 11769:Euclid's theorem 11739: 11738: 11656:Commensurability 11617:Axiomatic system 11565:Angle trisection 11530: 11520: 11482: 11472: 11462: 11452: 11428: 11418: 11401: 10964: 10957: 10950: 10941: 10940: 10934: 10932: 10931: 10918: 10916: 10915: 10910:. Whistler Alley 10902: 10900: 10899: 10884: 10883: 10865: 10863: 10862: 10843: 10827: 10813: 10803: 10797: 10789: 10769: 10767: 10766: 10760: 10754:. Archived from 10753: 10735: 10725: 10715: 10698: 10688: 10652: 10651: 10641: 10621: 10615: 10614: 10598: 10585: 10579: 10578: 10545:(5): 3168–3177. 10528: 10522: 10521: 10519: 10517:10.1038/139062a0 10485: 10479: 10477: 10438: 10426: 10391: 10389: 10378: 10363: 10351: 10339: 10324: 10302: 10292: 10280: 10274: 10264: 10258: 10257: 10239: 10233: 10232: 10208: 10202: 10201: 10185: 10172: 10166: 10165: 10141: 10114: 10108: 10107: 10081: 10072: 10050: 10020: 10014: 10013: 10003: 9985: 9953: 9947: 9946: 9932: 9926: 9925: 9914: 9908: 9907: 9868: 9862: 9861: 9859: 9828:(20 June 1936). 9822: 9816: 9815: 9772: 9763: 9750: 9740: 9726: 9717: 9716: 9704: 9698: 9697: 9695: 9694: 9676: 9640: 9634: 9633: 9599: 9576: 9570: 9569: 9541: 9535: 9534: 9517:(3–4): 497–542. 9506: 9500: 9499: 9470:Stoll V (1876). 9467: 9458: 9457: 9455: 9431: 9425: 9424: 9404: 9395: 9385: 9379: 9364: 9358: 9357: 9341: 9328: 9317: 9316: 9298: 9287: 9286: 9268: 9262: 9261: 9244:(1–2): 137–152. 9233: 9224: 9223: 9204: 9195: 9194: 9182: 9176: 9175: 9159: 9148:–355, 496, 499. 9143: 9130: 9112: 9101: 9100: 9089:10.1109/7.532270 9075:(3): 1116–1124. 9064: 9055: 9054: 9039: 9033: 9032: 9021: 9010: 9004: 9003: 8982: 8973: 8967: 8957: 8945: 8939: 8938: 8936: 8910: 8895: 8894: 8883: 8877: 8876: 8861: 8852: 8851: 8836: 8830: 8829: 8815: 8798: 8792: 8791: 8762: 8756: 8755: 8752: 8739: 8730: 8729: 8705: 8699: 8698: 8680: 8674: 8673: 8652: 8643: 8642: 8621: 8615: 8614: 8601: 8592: 8591: 8570: 8561: 8560: 8557: 8546: 8537: 8536: 8508: 8489: 8488: 8475: 8466: 8465: 8443: 8437: 8436: 8416: 8397: 8396: 8364: 8355: 8354: 8336: 8327: 8326: 8310: 8300: 8294: 8293: 8281: 8271: 8265: 8264: 8231: 8220: 8219: 8202: 8193: 8192: 8153: 8138: 8137: 8129: 8104:Isodynamic point 8094:Apollonius point 8080:of a pathogenic 8066:packing problems 7953: 7934:Pierre de Fermat 7737:The Kiss Precise 7641: 7639: 7638: 7633: 7627: 7622: 7609: 7604: 7591: 7586: 7573: 7568: 7550: 7549: 7540: 7539: 7527: 7526: 7514: 7513: 7501: 7500: 7366: 7343: 7320: 7297: 7274: 7251: 7228: 7205: 7182: 7159: 7117: 7005: 7001: 6997: 6989: 6985: 6981: 6977: 6966: 6960: 6954: 6947: 6945: 6944: 6939: 6931: 6930: 6921: 6918: 6910: 6909: 6900: 6899: 6894: 6873: 6872: 6860: 6859: 6847: 6846: 6815: 6811: 6773: 6764: 6755: 6743: 6732: 6726: 6715: 6704: 6698: 6687: 6677: 6667: 6661: 6651: 6625: 6619: 6612: 6602: 6589: 6572: 6555: 6549: 6543: 6537: 6527:is contained in 6526: 6520: 6514: 6505: 6499: 6493: 6487: 6481: 6475: 6466: 6460: 6449: 6443: 6437: 6427: 6421: 6415: 6409: 6400: 6391: 6385: 6375: 6369: 6360: 6351: 6337: 6326: 6318: 6312: 6300: 6293: 6282: 6276: 6272: 6268: 6262: 6256: 6252: 6246: 6240: 6234: 6228: 6222: 6206: 6200: 6194: 6188: 6178: 6176: 6175: 6170: 6159: 6156: 6151: 6148: 6140: 6139: 6134: 6082: 6076: 6070: 6056: 6052: 6046: 6042: 6032: 6026: 6020: 6011: 5998: 5992: 5985:Bézout's theorem 5982: 5976: 5970: 5961: 5952: 5946: 5940: 5934: 5925: 5921: 5917: 5910: 5888: 5886: 5885: 5880: 5875: 5874: 5865: 5864: 5852: 5851: 5824: 5823: 5785: 5775: 5773: 5772: 5767: 5749: 5747: 5746: 5741: 5717: 5715: 5714: 5709: 5675: 5673: 5672: 5667: 5628: 5614: 5605: 5596: 5592: 5586: 5576: 5559: 5557: 5556: 5551: 5546: 5545: 5540: 5480: 5479: 5440: 5439: 5412: 5411: 5396: 5395: 5390: 5347: 5340: 5334: 5328: 5213: 5211: 5210: 5205: 5203: 5198: 5197: 5196: 5187: 5186: 5176: 5171: 5166: 5165: 5164: 5155: 5154: 5144: 5139: 5134: 5133: 5132: 5123: 5122: 5112: 5107: 5102: 5101: 5100: 5091: 5090: 5080: 4468: 4466: 4465: 4460: 4455: 4451: 4448: 4437: 4426: 4425: 4384: 4382: 4381: 4376: 4371: 4370: 4358: 4356: 4355: 4354: 4353: 4337: 4336: 4335: 4321: 4319: 4314: 4313: 4299: 4294: 4293: 4273: 4268: 4267: 4256: 4155: 4109: 3991:complex analysis 3965: 3963: 3962: 3957: 3952: 3951: 3939: 3934: 3933: 3932: 3919: 3914: 3909: 3901: 3825:Circle inversion 3783: 3781: 3780: 3775: 3767: 3763: 3762: 3761: 3749: 3748: 3747: 3723: 3719: 3718: 3717: 3705: 3704: 3703: 3679: 3675: 3674: 3673: 3661: 3660: 3659: 3635: 3631: 3630: 3629: 3628: 3609: 3608: 3607: 3545: 3543: 3542: 3537: 3532: 3531: 3526: 3522: 3521: 3520: 3508: 3507: 3489: 3488: 3483: 3479: 3478: 3477: 3472: 3463: 3462: 3457: 3414: 3412: 3411: 3406: 3401: 3400: 3395: 3391: 3390: 3389: 3377: 3376: 3358: 3357: 3352: 3348: 3347: 3346: 3341: 3332: 3331: 3326: 3286: 3282: 3246: 3227: 3225: 3224: 3219: 3214: 3213: 3208: 3204: 3203: 3202: 3193: 3192: 3180: 3179: 3170: 3169: 3151: 3150: 3145: 3141: 3140: 3139: 3134: 3125: 3124: 3119: 3104: 3100: 3099: 3098: 3086: 3085: 3010: 3008: 3007: 3002: 2997: 2993: 2992: 2991: 2979: 2978: 2961: 2957: 2956: 2955: 2943: 2942: 2922: 2918: 2917: 2916: 2904: 2903: 2886: 2882: 2881: 2880: 2868: 2867: 2855: 2854: 2842: 2841: 2806: 2804: 2803: 2798: 2793: 2792: 2787: 2783: 2782: 2781: 2772: 2771: 2759: 2758: 2749: 2748: 2730: 2726: 2725: 2724: 2719: 2710: 2709: 2704: 2690: 2686: 2685: 2684: 2679: 2670: 2669: 2664: 2650: 2646: 2645: 2644: 2632: 2631: 2617: 2613: 2612: 2611: 2599: 2598: 2578: 2574: 2573: 2572: 2560: 2559: 2547: 2546: 2534: 2533: 2476: 2474: 2473: 2468: 2463: 2462: 2453: 2452: 2443: 2442: 2433: 2432: 2420: 2419: 2414: 2405: 2404: 2399: 2390: 2389: 2380: 2379: 2367: 2366: 2357: 2356: 2344: 2340: 2339: 2338: 2326: 2325: 2287:. The parameter 2189:circle inversion 2007: 2005: 2004: 1999: 1997: 1996: 1975: 1974: 1955: 1953: 1952: 1947: 1945: 1944: 1923: 1922: 1877: 1850: 1846: 1806: 1804: 1803: 1798: 1793: 1792: 1787: 1783: 1782: 1781: 1772: 1771: 1759: 1758: 1740: 1739: 1734: 1730: 1729: 1728: 1716: 1715: 1697: 1696: 1691: 1687: 1686: 1685: 1673: 1672: 1647: 1645: 1644: 1639: 1637: 1636: 1631: 1627: 1626: 1625: 1616: 1615: 1603: 1602: 1584: 1583: 1578: 1574: 1573: 1572: 1560: 1559: 1541: 1540: 1535: 1531: 1530: 1529: 1517: 1516: 1491: 1489: 1488: 1483: 1481: 1480: 1475: 1471: 1470: 1469: 1460: 1459: 1447: 1446: 1428: 1427: 1422: 1418: 1417: 1416: 1404: 1403: 1385: 1384: 1379: 1375: 1374: 1373: 1361: 1360: 1118:power of a point 907: 891: 871: 837: 817: 706:The solution of 692: 674: 630:Solution methods 577:circle inversion 569:power of a point 542: 535: 534: 506:). For example, 397: 367: 342: 317: 67: 60: 59: 54: 13024: 13023: 13019: 13018: 13017: 13015: 13014: 13013: 12984: 12983: 12982: 12977: 12976: 12975: 12974: 12973: 12966: 12953: 12909:Newton's cradle 12890: 12845: 12818: (student) 12816:William Whiston 12812: (student) 12768: 12749:Religious views 12710: 12625:Newton's method 12585:Newtonian fluid 12479:Bucket argument 12465: 12385: 12320: 12253: 12248: 12218: 12213: 12202: 12191: 12189: 12178: 12144:Arabian/Islamic 12132: 12121:numeral systems 12010: 11960: 11930: 11878:Heron's formula 11856: 11838: 11730: 11726:Triangle center 11716:Regular polygon 11593:and definitions 11592: 11586: 11548: 11528: 11518: 11480: 11470: 11460: 11450: 11426: 11416: 11399: 11365: 11336:Theon of Smyrna 10981: 10973: 10968: 10929: 10927: 10913: 10911: 10897: 10895: 10887: 10860: 10858: 10853: 10850: 10840: 10791: 10790: 10764: 10762: 10758: 10733: 10660: 10658:Further reading 10655: 10639:10.1.1.106.6518 10622: 10618: 10611: 10601:Springer-Verlag 10586: 10582: 10529: 10525: 10486: 10482: 10478: 10458:10.2307/2369532 10439: 10427: 10392: 10387: 10381:Reye T (1879). 10379: 10364: 10352: 10340: 10325: 10308: 10290: 10281: 10277: 10265: 10261: 10254: 10240: 10236: 10209: 10205: 10198: 10173: 10169: 10162: 10142: 10138: 10115: 10111: 10079: 10073: 10051: 10021: 10017: 9968:(11): 378–384. 9954: 9950: 9933: 9929: 9915: 9911: 9888:10.2307/2314247 9869: 9865: 9823: 9819: 9773: 9766: 9760:Wayback Machine 9741: 9727: 9720: 9714: 9705: 9701: 9692: 9690: 9677: 9657:10.2307/2975188 9641: 9637: 9606:10.2307/2695679 9577: 9573: 9542: 9538: 9507: 9503: 9468: 9461: 9432: 9428: 9405: 9398: 9386: 9382: 9365: 9361: 9354: 9329: 9320: 9313: 9299: 9290: 9283: 9269: 9265: 9234: 9227: 9205: 9198: 9183: 9179: 9160: 9156: 9131: 9127: 9113: 9104: 9065: 9058: 9040: 9036: 9022: 9011: 9007: 9000: 8983: 8976: 8955: 8946: 8942: 8911: 8898: 8884: 8880: 8862: 8855: 8837: 8833: 8816: 8808: 8799: 8795: 8788: 8763: 8759: 8753: 8740: 8733: 8726: 8708:Bold B (1982). 8706: 8702: 8695: 8681: 8677: 8670: 8653: 8646: 8639: 8622: 8618: 8602: 8595: 8588: 8571: 8564: 8558: 8547: 8540: 8525:10.2307/2690380 8509: 8492: 8476: 8469: 8444: 8440: 8417: 8400: 8365: 8358: 8351: 8337: 8330: 8323: 8301: 8297: 8290: 8272: 8268: 8261: 8232: 8223: 8203: 8196: 8173:10.2307/2315097 8154: 8141: 8130: 8117: 8113: 8090: 8054:Hans Rademacher 8001:multilateration 7972: 7956:Frederick Soddy 7948: 7922:Casey's theorem 7911:Kleinian groups 7868:Leibniz packing 7819: 7815: 7811: 7806: 7804:Generalizations 7794: 7793: 7790: 7788: 7786: 7784: 7782: 7780: 7778: 7776: 7774: 7769: 7768: 7765: 7763: 7761: 7759: 7757: 7755: 7753: 7751: 7749: 7743:of the circle. 7727:Frederick Soddy 7717: 7710: 7703: 7696: 7689: 7682: 7671: 7662: 7653: 7623: 7618: 7605: 7600: 7587: 7582: 7569: 7564: 7545: 7541: 7535: 7531: 7522: 7518: 7509: 7505: 7496: 7492: 7487: 7484: 7483: 7445:Soddy's circles 7432: 7375: 7135: 7130: 7033: 7027: 7022: 7013: 7003: 6999: 6991: 6987: 6983: 6979: 6976: 6968: 6962: 6956: 6952: 6926: 6922: 6917: 6905: 6901: 6895: 6890: 6889: 6868: 6864: 6855: 6851: 6842: 6838: 6824: 6821: 6820: 6813: 6810: 6809: 6798: 6797: 6786: 6785: 6775: 6772: 6766: 6763: 6757: 6754: 6748: 6742: 6734: 6728: 6725: 6717: 6714: 6706: 6700: 6697: 6689: 6683: 6673: 6663: 6653: 6627: 6621: 6614: 6604: 6597: 6591: 6584: 6578: 6571: 6557: 6551: 6545: 6539: 6536: 6528: 6522: 6516: 6513: 6507: 6501: 6495: 6489: 6483: 6477: 6471: 6462: 6459: 6451: 6445: 6439: 6429: 6423: 6417: 6411: 6408: 6402: 6399: 6393: 6387: 6381: 6371: 6368: 6362: 6359: 6353: 6350: 6342: 6336: 6328: 6324: 6314: 6302: 6296:Φ → 6295: 6292: 6284: 6278: 6274: 6270: 6264: 6258: 6254: 6253:, the fiber of 6248: 6242: 6236: 6230: 6224: 6221: 6212: 6207:means that the 6202: 6196: 6190: 6183: 6155: 6147: 6135: 6130: 6129: 6094: 6091: 6090: 6078: 6072: 6069: 6061: 6054: 6048: 6044: 6034: 6028: 6022: 6019: 6013: 6010: 6004: 5994: 5988: 5978: 5972: 5969: 5963: 5960: 5954: 5948: 5942: 5936: 5930: 5923: 5919: 5913: 5905: 5870: 5866: 5860: 5856: 5847: 5843: 5819: 5815: 5798: 5795: 5794: 5781: 5755: 5752: 5751: 5729: 5726: 5725: 5682: 5679: 5678: 5637: 5634: 5633: 5624: 5617:circular points 5615:are called the 5613: 5607: 5604: 5598: 5594: 5588: 5584: 5578: 5574: 5568: 5541: 5536: 5535: 5475: 5471: 5435: 5431: 5407: 5403: 5391: 5386: 5385: 5356: 5353: 5352: 5343: 5336: 5330: 5324: 5318:complex numbers 5302: 5291: 5282: 5273: 5266: 5259: 5252: 5245: 5234: 5223: 5192: 5188: 5182: 5178: 5177: 5175: 5160: 5156: 5150: 5146: 5145: 5143: 5128: 5124: 5118: 5114: 5113: 5111: 5096: 5092: 5086: 5082: 5081: 5079: 5077: 5074: 5073: 5068: 5058: 5051: 5044: 5037: 5030: 5023: 5016: 5009: 5002: 4995: 4988: 4950: 4943: 4936: 4925: 4918: 4903: 4896: 4885: 4878: 4871: 4864: 4853: 4846: 4839: 4832: 4825: 4807: 4800: 4793: 4776: 4769: 4762: 4755: 4748: 4742:that contained 4741: 4734: 4727: 4720: 4713: 4706: 4699: 4689: 4682: 4675: 4668: 4661: 4654: 4647: 4640: 4613: 4592: 4555: 4535: 4489: 4482: 4438: 4427: 4424: 4420: 4400: 4397: 4396: 4366: 4362: 4343: 4342: 4338: 4331: 4330: 4326: 4322: 4315: 4309: 4308: 4295: 4283: 4282: 4269: 4263: 4262: 4257: 4255: 4241: 4238: 4237: 4228: 4221: 4214: 4207: 4200: 4193: 4186: 4179: 4164: 4154: 4145: 4137: 4136:equals the sum 4135: 4118: 4108: 4099: 4091: 4089: 4055: 4018: 3981:is the same as 3947: 3943: 3928: 3924: 3920: 3918: 3902: 3900: 3898: 3895: 3894: 3801: 3757: 3753: 3737: 3736: 3732: 3731: 3727: 3713: 3709: 3693: 3692: 3688: 3687: 3683: 3669: 3665: 3649: 3648: 3644: 3643: 3639: 3618: 3617: 3613: 3597: 3596: 3592: 3591: 3587: 3585: 3582: 3581: 3576: 3569: 3562: 3555: 3527: 3516: 3512: 3503: 3499: 3498: 3494: 3493: 3484: 3473: 3468: 3467: 3458: 3453: 3452: 3451: 3447: 3446: 3444: 3441: 3440: 3431: 3424: 3396: 3385: 3381: 3372: 3368: 3367: 3363: 3362: 3353: 3342: 3337: 3336: 3327: 3322: 3321: 3320: 3316: 3315: 3313: 3310: 3309: 3300: 3293: 3284: 3280: 3278: 3271: 3264: 3257: 3245: 3238: 3232: 3209: 3198: 3194: 3188: 3184: 3175: 3171: 3165: 3161: 3160: 3156: 3155: 3146: 3135: 3130: 3129: 3120: 3115: 3114: 3113: 3109: 3108: 3094: 3090: 3081: 3077: 3076: 3072: 3064: 3061: 3060: 3055: 3048: 3041: 3034: 3027: 3020: 2987: 2983: 2974: 2970: 2969: 2965: 2951: 2947: 2938: 2934: 2933: 2929: 2912: 2908: 2899: 2895: 2894: 2890: 2876: 2872: 2863: 2859: 2850: 2846: 2837: 2833: 2832: 2828: 2826: 2823: 2822: 2788: 2777: 2773: 2767: 2763: 2754: 2750: 2744: 2740: 2739: 2735: 2734: 2720: 2715: 2714: 2705: 2700: 2699: 2698: 2694: 2680: 2675: 2674: 2665: 2660: 2659: 2658: 2654: 2640: 2636: 2627: 2623: 2622: 2618: 2607: 2603: 2594: 2590: 2589: 2585: 2568: 2564: 2555: 2551: 2542: 2538: 2529: 2525: 2524: 2520: 2518: 2515: 2514: 2509: 2502: 2458: 2454: 2448: 2444: 2438: 2434: 2428: 2424: 2415: 2410: 2409: 2400: 2395: 2394: 2385: 2381: 2375: 2371: 2362: 2358: 2352: 2348: 2334: 2330: 2321: 2317: 2316: 2312: 2310: 2307: 2306: 2275:represents the 2258: 2249: 2228: 2219: 2197: 2175: 2166: 2157: 2148: 2140:, then so is (− 2139: 2130: 2121: 2112: 2099: 2090: 2083: 2076: 2066: 2057: 2048: 2035: 1992: 1988: 1970: 1966: 1964: 1961: 1960: 1940: 1936: 1918: 1914: 1912: 1909: 1908: 1904: 1895: 1886: 1876: 1867: 1859: 1849:(2 × 2 × 2 = 8) 1848: 1844: 1833:right-hand side 1830: 1823: 1816: 1788: 1777: 1773: 1767: 1763: 1754: 1750: 1749: 1745: 1744: 1735: 1724: 1720: 1711: 1707: 1706: 1702: 1701: 1692: 1681: 1677: 1668: 1664: 1663: 1659: 1658: 1656: 1653: 1652: 1632: 1621: 1617: 1611: 1607: 1598: 1594: 1593: 1589: 1588: 1579: 1568: 1564: 1555: 1551: 1550: 1546: 1545: 1536: 1525: 1521: 1512: 1508: 1507: 1503: 1502: 1500: 1497: 1496: 1476: 1465: 1461: 1455: 1451: 1442: 1438: 1437: 1433: 1432: 1423: 1412: 1408: 1399: 1395: 1394: 1390: 1389: 1380: 1369: 1365: 1356: 1352: 1351: 1347: 1346: 1344: 1341: 1340: 1335: 1326: 1317: 1301: 1292: 1285: 1278: 1271: 1262: 1253: 1246: 1239: 1232: 1225: 1218: 1197: 1130:angle bisectors 1064: 1016:directrix lines 1009: 1002: 941: 934: 927: 916: 906: 899: 893: 890: 881: 873: 870: 861: 853: 851: 844: 836: 827: 819: 816: 807: 799: 797: 790: 783: 774: 765: 752: 745: 734: 727: 720: 701: 691: 682: 676: 673: 664: 658: 656: 649: 637: 632: 581:Julius Petersen 443: 419:solutions below 416: 406: 396: 389: 382: 375: 369: 366: 357: 350: 344: 341: 332: 325: 319: 316: 307: 300: 294: 292: 285: 278: 271: 181: 97:constructions. 82:3 in 2 parts). 52: 17: 12: 11: 5: 13022: 13012: 13011: 13006: 13001: 12996: 12979: 12978: 12965: 12964: 12962: 12961: 12959: 12955: 12954: 12952: 12951: 12946: 12941: 12936: 12931: 12926: 12921: 12916: 12911: 12906: 12900: 12898: 12892: 12891: 12889: 12888: 12881: 12874: 12865: 12855: 12853: 12847: 12846: 12844: 12843: 12842: (friend) 12837: 12836: (friend) 12831: 12830: (friend) 12825: 12819: 12813: 12807: 12801: 12800: (mentor) 12798:William Clarke 12795: 12789: 12783: 12776: 12774: 12770: 12769: 12767: 12766: 12761: 12756: 12754:Occult studies 12751: 12746: 12741: 12736: 12731: 12725: 12718: 12716: 12712: 12711: 12709: 12708: 12707: 12706: 12696: 12691: 12686: 12681: 12676: 12671: 12670: 12669: 12659: 12654: 12649: 12644: 12639: 12637:Newton fractal 12634: 12633: 12632: 12622: 12617: 12612: 12607: 12602: 12597: 12592: 12587: 12582: 12577: 12572: 12567: 12565:Newton's rings 12562: 12557: 12556: 12555: 12550: 12540: 12535: 12534: 12533: 12523: 12518: 12513: 12512: 12511: 12506: 12501: 12491: 12486: 12481: 12475: 12473: 12467: 12466: 12464: 12463: 12458: 12453: 12451:Newton's metal 12448: 12443: 12438: 12433: 12432: 12431: 12424:Newton polygon 12421: 12416: 12411: 12406: 12405: 12404: 12393: 12391: 12387: 12386: 12384: 12383: 12375: 12367: 12358:" (1713; 12352: 12344: 12337: 12328: 12326: 12325:Other writings 12322: 12321: 12319: 12318: 12310: 12302: 12294: 12286: 12278: 12270: 12261: 12259: 12255: 12254: 12247: 12246: 12239: 12232: 12224: 12215: 12214: 12187: 12184: 12183: 12180: 12179: 12177: 12176: 12171: 12166: 12161: 12156: 12151: 12146: 12140: 12138: 12137:Other cultures 12134: 12133: 12131: 12130: 12129: 12128: 12118: 12117: 12116: 12106: 12105: 12104: 12094: 12093: 12092: 12082: 12081: 12080: 12070: 12069: 12068: 12058: 12057: 12056: 12046: 12045: 12044: 12034: 12033: 12032: 12018: 12016: 12012: 12011: 12009: 12008: 12003: 11998: 11993: 11988: 11986:Greek numerals 11983: 11981:Attic numerals 11978: 11972: 11966: 11962: 11961: 11959: 11958: 11953: 11948: 11942: 11940: 11936: 11935: 11932: 11931: 11929: 11928: 11923: 11918: 11913: 11908: 11900: 11895: 11890: 11885: 11880: 11875: 11870: 11864: 11862: 11858: 11857: 11855: 11854: 11848: 11846: 11840: 11839: 11837: 11836: 11831: 11826: 11821: 11816: 11811: 11809:Law of cosines 11806: 11801: 11796: 11791: 11786: 11781: 11776: 11771: 11766: 11761: 11756: 11750: 11748: 11736: 11732: 11731: 11729: 11728: 11723: 11718: 11713: 11708: 11703: 11701:Platonic solid 11698: 11693: 11688: 11683: 11681:Greek numerals 11678: 11673: 11668: 11663: 11658: 11653: 11648: 11647: 11646: 11641: 11631: 11626: 11625: 11624: 11614: 11613: 11612: 11607: 11596: 11594: 11588: 11587: 11585: 11584: 11579: 11578: 11577: 11572: 11567: 11556: 11554: 11550: 11549: 11547: 11546: 11539: 11532: 11522: 11512: 11509:Planisphaerium 11505: 11498: 11491: 11484: 11474: 11464: 11454: 11444: 11437: 11430: 11420: 11410: 11403: 11393: 11386: 11381: 11373: 11371: 11367: 11366: 11364: 11363: 11358: 11353: 11348: 11343: 11338: 11333: 11328: 11323: 11318: 11313: 11308: 11303: 11298: 11293: 11288: 11283: 11278: 11273: 11268: 11263: 11258: 11253: 11248: 11243: 11238: 11233: 11228: 11223: 11218: 11213: 11208: 11203: 11198: 11193: 11188: 11183: 11178: 11173: 11168: 11163: 11158: 11153: 11148: 11143: 11138: 11133: 11128: 11123: 11118: 11113: 11108: 11103: 11098: 11093: 11088: 11083: 11078: 11073: 11068: 11063: 11058: 11053: 11048: 11043: 11038: 11033: 11028: 11023: 11018: 11013: 11008: 11003: 10998: 10993: 10987: 10985: 10979:Mathematicians 10975: 10974: 10967: 10966: 10959: 10952: 10944: 10936: 10935: 10919: 10906:Kunkel, Paul. 10903: 10885: 10866: 10849: 10848:External links 10846: 10845: 10844: 10838: 10814: 10805: 10770: 10726: 10716: 10699: 10679:(2): 303–322. 10659: 10656: 10654: 10653: 10632:(2): 101–114. 10616: 10609: 10580: 10523: 10480: 10275: 10259: 10252: 10234: 10203: 10196: 10167: 10160: 10136: 10109: 10090:(4): 691–721. 10063:(2): 170–174. 10015: 9948: 9927: 9909: 9882:(6): 627–640. 9863: 9842:(3477): 1021. 9817: 9764: 9718: 9699: 9651:(4): 319–329. 9635: 9571: 9536: 9501: 9482:(4): 613–632. 9459: 9426: 9396: 9380: 9376:978-1107602724 9359: 9352: 9318: 9311: 9288: 9281: 9263: 9225: 9196: 9177: 9154: 9125: 9102: 9056: 9034: 9005: 8998: 8990:Werke, 4. Band 8974: 8940: 8927:(2): 725–744. 8896: 8878: 8853: 8831: 8793: 8786: 8757: 8731: 8724: 8700: 8693: 8675: 8668: 8644: 8637: 8616: 8593: 8586: 8562: 8538: 8490: 8467: 8438: 8398: 8356: 8349: 8328: 8321: 8295: 8288: 8266: 8259: 8221: 8194: 8139: 8114: 8112: 8109: 8108: 8107: 8101: 8096: 8089: 8086: 8013:speed of sound 7971: 7968: 7849:spherical caps 7817: 7813: 7809: 7805: 7802: 7771: 7770: 7746: 7745: 7715: 7708: 7701: 7694: 7687: 7680: 7667: 7658: 7649: 7643: 7642: 7631: 7626: 7621: 7617: 7613: 7608: 7603: 7599: 7595: 7590: 7585: 7581: 7577: 7572: 7567: 7563: 7559: 7556: 7553: 7548: 7544: 7538: 7534: 7530: 7525: 7521: 7517: 7512: 7508: 7504: 7499: 7495: 7491: 7477:René Descartes 7431: 7428: 7374: 7371: 7368: 7367: 7360: 7357: 7354: 7349: 7345: 7344: 7337: 7334: 7331: 7326: 7322: 7321: 7314: 7311: 7308: 7303: 7299: 7298: 7291: 7288: 7285: 7280: 7276: 7275: 7268: 7265: 7262: 7257: 7253: 7252: 7245: 7242: 7239: 7234: 7230: 7229: 7222: 7219: 7216: 7211: 7207: 7206: 7199: 7196: 7193: 7188: 7184: 7183: 7176: 7173: 7170: 7165: 7161: 7160: 7153: 7150: 7147: 7142: 7138: 7137: 7132: 7127: 7126:Given Elements 7124: 7121: 7108:limiting cases 7059:Some of these 7029:Main article: 7026: 7023: 7021: 7018: 7012: 7009: 6986:has dimension 6972: 6949: 6948: 6937: 6934: 6929: 6925: 6916: 6913: 6908: 6904: 6898: 6893: 6888: 6885: 6882: 6879: 6876: 6871: 6867: 6863: 6858: 6854: 6850: 6845: 6841: 6837: 6834: 6831: 6828: 6807: 6803: 6795: 6791: 6783: 6779: 6770: 6761: 6752: 6738: 6721: 6710: 6693: 6595: 6582: 6567: 6532: 6515:is tangent to 6511: 6455: 6406: 6397: 6366: 6357: 6346: 6332: 6288: 6217: 6180: 6179: 6168: 6165: 6162: 6157: at 6154: 6146: 6143: 6138: 6133: 6128: 6125: 6122: 6119: 6116: 6113: 6110: 6107: 6104: 6101: 6098: 6065: 6017: 6008: 5967: 5958: 5895:smooth circles 5890: 5889: 5878: 5873: 5869: 5863: 5859: 5855: 5850: 5846: 5842: 5839: 5836: 5833: 5830: 5827: 5822: 5818: 5814: 5811: 5808: 5805: 5802: 5778: 5777: 5765: 5762: 5759: 5739: 5736: 5733: 5719: 5718: 5707: 5704: 5701: 5698: 5695: 5692: 5689: 5686: 5676: 5665: 5662: 5659: 5656: 5653: 5650: 5647: 5644: 5641: 5611: 5602: 5597:. The points 5582: 5572: 5561: 5560: 5549: 5544: 5539: 5534: 5531: 5528: 5525: 5522: 5519: 5516: 5513: 5510: 5507: 5504: 5501: 5498: 5495: 5492: 5489: 5486: 5483: 5478: 5474: 5470: 5467: 5464: 5461: 5458: 5455: 5452: 5449: 5446: 5443: 5438: 5434: 5430: 5427: 5424: 5421: 5418: 5415: 5410: 5406: 5402: 5399: 5394: 5389: 5384: 5381: 5378: 5375: 5372: 5369: 5366: 5363: 5360: 5301: 5298: 5287: 5278: 5271: 5264: 5257: 5250: 5243: 5232: 5221: 5215: 5214: 5201: 5195: 5191: 5185: 5181: 5174: 5169: 5163: 5159: 5153: 5149: 5142: 5137: 5131: 5127: 5121: 5117: 5110: 5105: 5099: 5095: 5089: 5085: 5066: 5056: 5049: 5042: 5035: 5028: 5021: 5014: 5007: 5000: 4993: 4986: 4964:radical center 4948: 4941: 4934: 4923: 4916: 4908:, the pole of 4901: 4894: 4883: 4876: 4869: 4862: 4851: 4844: 4837: 4830: 4823: 4805: 4798: 4791: 4779:radical center 4774: 4767: 4760: 4753: 4746: 4739: 4732: 4725: 4718: 4711: 4704: 4697: 4687: 4680: 4673: 4666: 4659: 4652: 4645: 4638: 4612: 4609: 4591: 4588: 4554: 4551: 4534: 4531: 4487: 4480: 4470: 4469: 4458: 4454: 4447: 4444: 4441: 4436: 4433: 4430: 4423: 4419: 4416: 4413: 4410: 4407: 4404: 4386: 4385: 4374: 4369: 4365: 4361: 4352: 4349: 4346: 4341: 4334: 4329: 4325: 4318: 4312: 4307: 4303: 4298: 4292: 4289: 4286: 4281: 4277: 4272: 4266: 4261: 4254: 4251: 4248: 4245: 4231:law of cosines 4226: 4219: 4212: 4205: 4198: 4191: 4184: 4177: 4160: 4150: 4141: 4131: 4114: 4104: 4095: 4085: 4054: 4051: 4042:radical center 4038:radical circle 4017: 4014: 4006:Riemann sphere 3995:Riemann sphere 3967: 3966: 3955: 3950: 3946: 3942: 3937: 3931: 3927: 3923: 3917: 3912: 3908: 3905: 3882:to the center 3800: 3797: 3789:linear algebra 3785: 3784: 3773: 3770: 3766: 3760: 3756: 3752: 3746: 3743: 3740: 3735: 3730: 3726: 3722: 3716: 3712: 3708: 3702: 3699: 3696: 3691: 3686: 3682: 3678: 3672: 3668: 3664: 3658: 3655: 3652: 3647: 3642: 3638: 3634: 3627: 3624: 3621: 3616: 3612: 3606: 3603: 3600: 3595: 3590: 3574: 3567: 3560: 3553: 3547: 3546: 3535: 3530: 3525: 3519: 3515: 3511: 3506: 3502: 3497: 3492: 3487: 3482: 3476: 3471: 3466: 3461: 3456: 3450: 3429: 3422: 3416: 3415: 3404: 3399: 3394: 3388: 3384: 3380: 3375: 3371: 3366: 3361: 3356: 3351: 3345: 3340: 3335: 3330: 3325: 3319: 3298: 3291: 3276: 3269: 3262: 3255: 3249:Euclidean norm 3243: 3236: 3229: 3228: 3217: 3212: 3207: 3201: 3197: 3191: 3187: 3183: 3178: 3174: 3168: 3164: 3159: 3154: 3149: 3144: 3138: 3133: 3128: 3123: 3118: 3112: 3107: 3103: 3097: 3093: 3089: 3084: 3080: 3075: 3071: 3068: 3053: 3046: 3039: 3032: 3025: 3018: 3012: 3011: 3000: 2996: 2990: 2986: 2982: 2977: 2973: 2968: 2964: 2960: 2954: 2950: 2946: 2941: 2937: 2932: 2928: 2925: 2921: 2915: 2911: 2907: 2902: 2898: 2893: 2889: 2885: 2879: 2875: 2871: 2866: 2862: 2858: 2853: 2849: 2845: 2840: 2836: 2831: 2808: 2807: 2796: 2791: 2786: 2780: 2776: 2770: 2766: 2762: 2757: 2753: 2747: 2743: 2738: 2733: 2729: 2723: 2718: 2713: 2708: 2703: 2697: 2693: 2689: 2683: 2678: 2673: 2668: 2663: 2657: 2653: 2649: 2643: 2639: 2635: 2630: 2626: 2621: 2616: 2610: 2606: 2602: 2597: 2593: 2588: 2584: 2581: 2577: 2571: 2567: 2563: 2558: 2554: 2550: 2545: 2541: 2537: 2532: 2528: 2523: 2507: 2500: 2478: 2477: 2466: 2461: 2457: 2451: 2447: 2441: 2437: 2431: 2427: 2423: 2418: 2413: 2408: 2403: 2398: 2393: 2388: 2384: 2378: 2374: 2370: 2365: 2361: 2355: 2351: 2347: 2343: 2337: 2333: 2329: 2324: 2320: 2315: 2254: 2245: 2224: 2215: 2196: 2193: 2171: 2162: 2153: 2144: 2135: 2126: 2117: 2108: 2095: 2088: 2081: 2074: 2062: 2053: 2044: 2031: 2009: 2008: 1995: 1991: 1987: 1984: 1981: 1978: 1973: 1969: 1957: 1956: 1943: 1939: 1935: 1932: 1929: 1926: 1921: 1917: 1900: 1891: 1882: 1872: 1863: 1828: 1821: 1814: 1808: 1807: 1796: 1791: 1786: 1780: 1776: 1770: 1766: 1762: 1757: 1753: 1748: 1743: 1738: 1733: 1727: 1723: 1719: 1714: 1710: 1705: 1700: 1695: 1690: 1684: 1680: 1676: 1671: 1667: 1662: 1649: 1648: 1635: 1630: 1624: 1620: 1614: 1610: 1606: 1601: 1597: 1592: 1587: 1582: 1577: 1571: 1567: 1563: 1558: 1554: 1549: 1544: 1539: 1534: 1528: 1524: 1520: 1515: 1511: 1506: 1493: 1492: 1479: 1474: 1468: 1464: 1458: 1454: 1450: 1445: 1441: 1436: 1431: 1426: 1421: 1415: 1411: 1407: 1402: 1398: 1393: 1388: 1383: 1378: 1372: 1368: 1364: 1359: 1355: 1350: 1331: 1322: 1313: 1297: 1290: 1283: 1276: 1267: 1258: 1251: 1244: 1237: 1230: 1223: 1216: 1196: 1193: 1063: 1060: 1007: 1000: 939: 932: 925: 912: 904: 897: 886: 877: 866: 857: 849: 842: 832: 823: 812: 803: 795: 788: 779: 770: 761: 750: 743: 732: 725: 718: 697: 687: 680: 669: 662: 654: 647: 636: 633: 631: 628: 608:Leonhard Euler 600:René Descartes 559:(1811) and of 550:De contactibus 546:De tactionibus 523:François Viète 504:conic sections 451:François Viète 442: 439: 429:, and also in 412: 402: 394: 387: 380: 373: 362: 355: 348: 337: 330: 323: 312: 305: 298: 290: 283: 276: 267: 180: 177: 149:René Descartes 103:limiting cases 99:François Viète 15: 9: 6: 4: 3: 2: 13021: 13010: 13007: 13005: 13002: 13000: 12997: 12995: 12992: 12991: 12989: 12972: 12968: 12960: 12956: 12950: 12947: 12945: 12942: 12940: 12937: 12935: 12932: 12930: 12927: 12925: 12922: 12920: 12917: 12915: 12912: 12910: 12907: 12905: 12904:Newton (unit) 12902: 12901: 12899: 12897: 12893: 12887: 12886: 12882: 12880: 12879: 12875: 12872: 12870: 12866: 12863: 12861: 12857: 12856: 12854: 12852: 12848: 12841: 12838: 12835: 12834:William Jones 12832: 12829: 12826: 12823: 12820: 12817: 12814: 12811: 12808: 12806: (tutor) 12805: 12802: 12799: 12796: 12793: 12790: 12787: 12786:John Conduitt 12784: 12782: (niece) 12781: 12778: 12777: 12775: 12771: 12765: 12762: 12760: 12757: 12755: 12752: 12750: 12747: 12745: 12742: 12740: 12737: 12735: 12732: 12729: 12728:Cranbury Park 12726: 12723: 12720: 12719: 12717: 12715:Personal life 12713: 12705: 12702: 12701: 12700: 12697: 12695: 12692: 12690: 12687: 12685: 12682: 12680: 12677: 12675: 12672: 12668: 12665: 12664: 12663: 12662:Newton number 12660: 12658: 12655: 12653: 12650: 12648: 12645: 12643: 12640: 12638: 12635: 12631: 12628: 12627: 12626: 12623: 12621: 12618: 12616: 12613: 12611: 12608: 12606: 12603: 12601: 12598: 12596: 12593: 12591: 12588: 12586: 12583: 12581: 12578: 12576: 12573: 12571: 12568: 12566: 12563: 12561: 12558: 12554: 12551: 12549: 12546: 12545: 12544: 12541: 12539: 12536: 12532: 12531:Kepler's laws 12529: 12528: 12527: 12524: 12522: 12519: 12517: 12514: 12510: 12507: 12505: 12504:parameterized 12502: 12500: 12497: 12496: 12495: 12492: 12490: 12487: 12485: 12482: 12480: 12477: 12476: 12474: 12472: 12468: 12462: 12459: 12457: 12454: 12452: 12449: 12447: 12444: 12442: 12439: 12437: 12434: 12430: 12427: 12426: 12425: 12422: 12420: 12417: 12415: 12412: 12410: 12407: 12403: 12400: 12399: 12398: 12395: 12394: 12392: 12390:Contributions 12388: 12381: 12380: 12376: 12373: 12372: 12368: 12365: 12363: 12357: 12353: 12350: 12349: 12345: 12343:" (1675) 12342: 12338: 12335: 12334: 12330: 12329: 12327: 12323: 12316: 12315: 12311: 12308: 12307: 12303: 12300: 12299: 12295: 12292: 12291: 12287: 12284: 12283: 12279: 12276: 12275: 12271: 12268: 12267: 12263: 12262: 12260: 12256: 12252: 12245: 12240: 12238: 12233: 12231: 12226: 12225: 12222: 12212: 12211: 12206: 12199: 12198: 12185: 12175: 12172: 12170: 12167: 12165: 12162: 12160: 12157: 12155: 12152: 12150: 12147: 12145: 12142: 12141: 12139: 12135: 12127: 12124: 12123: 12122: 12119: 12115: 12112: 12111: 12110: 12107: 12103: 12100: 12099: 12098: 12095: 12091: 12088: 12087: 12086: 12083: 12079: 12076: 12075: 12074: 12071: 12067: 12064: 12063: 12062: 12059: 12055: 12052: 12051: 12050: 12047: 12043: 12040: 12039: 12038: 12035: 12031: 12027: 12026: 12025: 12024: 12020: 12019: 12017: 12013: 12007: 12004: 12002: 11999: 11997: 11994: 11992: 11989: 11987: 11984: 11982: 11979: 11977: 11974: 11973: 11970: 11967: 11963: 11957: 11954: 11952: 11949: 11947: 11944: 11943: 11941: 11937: 11927: 11924: 11922: 11919: 11917: 11914: 11912: 11909: 11907: 11906: 11901: 11899: 11896: 11894: 11891: 11889: 11886: 11884: 11881: 11879: 11876: 11874: 11871: 11869: 11866: 11865: 11863: 11859: 11853: 11850: 11849: 11847: 11845: 11841: 11835: 11832: 11830: 11827: 11825: 11822: 11820: 11817: 11815: 11814:Pons asinorum 11812: 11810: 11807: 11805: 11802: 11800: 11797: 11795: 11792: 11790: 11787: 11785: 11784:Hinge theorem 11782: 11780: 11777: 11775: 11772: 11770: 11767: 11765: 11762: 11760: 11757: 11755: 11752: 11751: 11749: 11747: 11746: 11740: 11737: 11733: 11727: 11724: 11722: 11719: 11717: 11714: 11712: 11709: 11707: 11704: 11702: 11699: 11697: 11694: 11692: 11689: 11687: 11684: 11682: 11679: 11677: 11674: 11672: 11669: 11667: 11664: 11662: 11659: 11657: 11654: 11652: 11649: 11645: 11642: 11640: 11637: 11636: 11635: 11632: 11630: 11627: 11623: 11620: 11619: 11618: 11615: 11611: 11608: 11606: 11603: 11602: 11601: 11598: 11597: 11595: 11589: 11583: 11580: 11576: 11573: 11571: 11568: 11566: 11563: 11562: 11561: 11558: 11557: 11555: 11551: 11545: 11544: 11540: 11538: 11537: 11533: 11531: 11527: 11523: 11521: 11517: 11513: 11511: 11510: 11506: 11504: 11503: 11499: 11497: 11496: 11492: 11490: 11489: 11485: 11483: 11479: 11475: 11473: 11469: 11465: 11463: 11459: 11455: 11453: 11451:(Aristarchus) 11449: 11445: 11443: 11442: 11438: 11436: 11435: 11431: 11429: 11425: 11421: 11419: 11415: 11411: 11409: 11408: 11404: 11402: 11398: 11394: 11392: 11391: 11387: 11385: 11382: 11380: 11379: 11375: 11374: 11372: 11368: 11362: 11359: 11357: 11356:Zeno of Sidon 11354: 11352: 11349: 11347: 11344: 11342: 11339: 11337: 11334: 11332: 11329: 11327: 11324: 11322: 11319: 11317: 11314: 11312: 11309: 11307: 11304: 11302: 11299: 11297: 11294: 11292: 11289: 11287: 11284: 11282: 11279: 11277: 11274: 11272: 11269: 11267: 11264: 11262: 11259: 11257: 11254: 11252: 11249: 11247: 11244: 11242: 11239: 11237: 11234: 11232: 11229: 11227: 11224: 11222: 11219: 11217: 11214: 11212: 11209: 11207: 11204: 11202: 11199: 11197: 11194: 11192: 11189: 11187: 11184: 11182: 11179: 11177: 11174: 11172: 11169: 11167: 11164: 11162: 11159: 11157: 11154: 11152: 11149: 11147: 11144: 11142: 11139: 11137: 11134: 11132: 11129: 11127: 11124: 11122: 11119: 11117: 11114: 11112: 11109: 11107: 11104: 11102: 11099: 11097: 11094: 11092: 11089: 11087: 11084: 11082: 11079: 11077: 11074: 11072: 11069: 11067: 11064: 11062: 11059: 11057: 11054: 11052: 11049: 11047: 11044: 11042: 11039: 11037: 11034: 11032: 11029: 11027: 11024: 11022: 11019: 11017: 11014: 11012: 11009: 11007: 11004: 11002: 10999: 10997: 10994: 10992: 10989: 10988: 10986: 10984: 10980: 10976: 10972: 10965: 10960: 10958: 10953: 10951: 10946: 10945: 10942: 10938: 10925: 10920: 10909: 10904: 10894: 10890: 10886: 10881: 10880: 10875: 10872: 10867: 10856: 10852: 10851: 10841: 10839:0-14-011813-6 10835: 10831: 10826: 10825: 10819: 10815: 10811: 10806: 10801: 10795: 10787: 10783: 10779: 10775: 10771: 10761:on 2008-04-15 10757: 10752: 10747: 10743: 10739: 10732: 10727: 10723: 10717: 10713: 10709: 10705: 10700: 10696: 10692: 10687: 10682: 10678: 10674: 10670: 10666: 10662: 10661: 10649: 10645: 10640: 10635: 10631: 10627: 10620: 10612: 10606: 10602: 10597: 10596: 10590: 10584: 10576: 10572: 10568: 10564: 10560: 10556: 10552: 10548: 10544: 10540: 10539: 10534: 10527: 10518: 10513: 10509: 10505: 10501: 10497: 10496: 10491: 10484: 10475: 10471: 10467: 10463: 10459: 10455: 10451: 10447: 10443: 10436: 10432: 10424: 10420: 10416: 10412: 10409:(37): 51–57. 10408: 10404: 10400: 10396: 10386: 10385: 10376: 10373:(in French). 10372: 10368: 10362:(5): 409–410. 10361: 10358:(in French). 10357: 10349: 10346:(in French). 10345: 10337: 10334:(in French). 10333: 10329: 10322: 10318: 10317: 10312: 10306: 10300: 10296: 10289: 10285: 10279: 10272: 10268: 10263: 10255: 10253:1-4297-0050-5 10249: 10245: 10238: 10230: 10226: 10222: 10218: 10214: 10207: 10199: 10197:0-521-35253-3 10193: 10189: 10184: 10183: 10177: 10171: 10163: 10157: 10153: 10152: 10147: 10139: 10133: 10129: 10125: 10124: 10119: 10113: 10105: 10101: 10097: 10093: 10089: 10085: 10078: 10070: 10066: 10062: 10058: 10057: 10048: 10044: 10040: 10036: 10032: 10028: 10027: 10019: 10011: 10007: 10002: 9997: 9993: 9989: 9984: 9979: 9975: 9971: 9967: 9963: 9959: 9952: 9944: 9941:(in French). 9940: 9939: 9931: 9923: 9919: 9913: 9905: 9901: 9897: 9893: 9889: 9885: 9881: 9877: 9873: 9867: 9858: 9853: 9849: 9845: 9841: 9837: 9836: 9831: 9827: 9821: 9813: 9809: 9805: 9801: 9797: 9793: 9789: 9785: 9781: 9777: 9771: 9769: 9761: 9757: 9754: 9748: 9747: 9738: 9734: 9733: 9725: 9723: 9712: 9708: 9703: 9689: 9685: 9681: 9680:Weisstein, EW 9674: 9670: 9666: 9662: 9658: 9654: 9650: 9646: 9639: 9631: 9627: 9623: 9619: 9615: 9611: 9607: 9603: 9598: 9593: 9589: 9585: 9581: 9575: 9567: 9563: 9559: 9555: 9551: 9547: 9540: 9532: 9528: 9524: 9520: 9516: 9513:(in German). 9512: 9505: 9497: 9493: 9489: 9485: 9481: 9478:(in German). 9477: 9473: 9466: 9464: 9454: 9449: 9445: 9441: 9437: 9430: 9422: 9418: 9414: 9412: 9403: 9401: 9394: 9390: 9384: 9377: 9373: 9369: 9363: 9355: 9353:0-486-26530-7 9349: 9345: 9340: 9339: 9333: 9332:Ogilvy, C. S. 9327: 9325: 9323: 9314: 9308: 9304: 9297: 9295: 9293: 9284: 9282:0-8284-0098-9 9278: 9274: 9267: 9259: 9255: 9251: 9247: 9243: 9239: 9232: 9230: 9221: 9217: 9213: 9209: 9203: 9201: 9192: 9188: 9181: 9173: 9169: 9165: 9157: 9151: 9147: 9142: 9141: 9135: 9128: 9122: 9118: 9111: 9109: 9107: 9098: 9094: 9090: 9086: 9082: 9078: 9074: 9070: 9063: 9061: 9053:(6): 193–195. 9052: 9049:(in French). 9048: 9044: 9038: 9030: 9026: 9019: 9015: 9009: 9001: 8999:3-487-04636-9 8995: 8991: 8987: 8981: 8979: 8971: 8965: 8961: 8954: 8950: 8944: 8935: 8930: 8926: 8922: 8921: 8916: 8909: 8907: 8905: 8903: 8901: 8892: 8888: 8882: 8874: 8871:(in French). 8870: 8866: 8860: 8858: 8850:(3): 271–273. 8849: 8846:(in French). 8845: 8841: 8835: 8827: 8825: 8820: 8813: 8806: 8802: 8797: 8789: 8787:0-471-54397-7 8783: 8779: 8775: 8771: 8767: 8761: 8750: 8749: 8744: 8738: 8736: 8727: 8725:0-486-24297-8 8721: 8717: 8713: 8712: 8704: 8696: 8694:0-19-510519-2 8690: 8686: 8679: 8671: 8665: 8661: 8657: 8651: 8649: 8640: 8638:0-521-08719-8 8634: 8630: 8626: 8620: 8612: 8611: 8606: 8600: 8598: 8589: 8587:0-521-08719-8 8583: 8579: 8575: 8569: 8567: 8555: 8551: 8545: 8543: 8534: 8530: 8526: 8522: 8519:(2): 97–103. 8518: 8514: 8507: 8505: 8503: 8501: 8499: 8497: 8495: 8486: 8485: 8480: 8474: 8472: 8463: 8459: 8455: 8451: 8450: 8442: 8434: 8430: 8426: 8422: 8415: 8413: 8411: 8409: 8407: 8405: 8403: 8394: 8390: 8386: 8382: 8378: 8374: 8370: 8363: 8361: 8352: 8346: 8342: 8335: 8333: 8324: 8318: 8314: 8309: 8308: 8299: 8291: 8285: 8280: 8279: 8270: 8262: 8256: 8252: 8248: 8244: 8240: 8236: 8230: 8228: 8226: 8217: 8213: 8212: 8207: 8201: 8199: 8190: 8186: 8182: 8178: 8174: 8170: 8166: 8162: 8158: 8152: 8150: 8148: 8146: 8144: 8135: 8128: 8126: 8124: 8122: 8120: 8115: 8106:of a triangle 8105: 8102: 8100: 8097: 8095: 8092: 8091: 8085: 8083: 8079: 8075: 8071: 8067: 8063: 8059: 8055: 8052:to construct 8051: 8047: 8043: 8039: 8035: 8031: 8030: 8024: 8022: 8018: 8014: 8010: 8006: 8002: 7998: 7994: 7990: 7986: 7981: 7977: 7967: 7965: 7961: 7957: 7951: 7946: 7942: 7937: 7935: 7929: 7927: 7923: 7919: 7914: 7912: 7908: 7904: 7900: 7896: 7892: 7888: 7884: 7881: 7877: 7873: 7869: 7865: 7860: 7858: 7857:Charles Dupin 7854: 7850: 7846: 7842: 7834: 7829: 7825: 7823: 7801: 7799: 7792: 7767: 7744: 7742: 7738: 7734: 7733: 7728: 7724: 7723:Jakob Steiner 7719: 7714: 7707: 7700: 7693: 7686: 7679: 7675: 7670: 7666: 7661: 7657: 7652: 7648: 7624: 7619: 7615: 7611: 7606: 7601: 7597: 7593: 7588: 7583: 7579: 7575: 7570: 7565: 7561: 7554: 7551: 7546: 7536: 7532: 7528: 7523: 7519: 7515: 7510: 7506: 7502: 7497: 7493: 7482: 7481: 7480: 7478: 7474: 7470: 7460: 7456: 7454: 7453:Steiner chain 7450: 7446: 7442: 7438: 7427: 7425: 7421: 7417: 7413: 7409: 7403: 7401: 7398:, as used in 7397: 7392: 7388: 7379: 7365: 7361: 7358: 7355: 7353: 7350: 7347: 7346: 7342: 7338: 7335: 7332: 7330: 7327: 7324: 7323: 7319: 7315: 7312: 7309: 7307: 7304: 7301: 7300: 7296: 7292: 7289: 7286: 7284: 7281: 7278: 7277: 7273: 7269: 7266: 7263: 7261: 7258: 7255: 7254: 7250: 7246: 7243: 7240: 7238: 7235: 7232: 7231: 7227: 7223: 7220: 7217: 7215: 7212: 7209: 7208: 7204: 7200: 7197: 7194: 7192: 7189: 7186: 7185: 7181: 7177: 7174: 7171: 7169: 7166: 7163: 7162: 7158: 7154: 7151: 7148: 7146: 7143: 7140: 7139: 7133: 7128: 7125: 7122: 7119: 7118: 7112: 7109: 7103: 7101: 7098:problem (the 7097: 7092: 7088: 7084: 7080: 7079: 7074: 7070: 7066: 7062: 7061:special cases 7057: 7055: 7051: 7047: 7043: 7039: 7032: 7020:Special cases 7017: 7008: 6995: 6975: 6971: 6965: 6959: 6935: 6927: 6923: 6914: 6911: 6906: 6896: 6883: 6877: 6874: 6869: 6865: 6861: 6856: 6852: 6848: 6843: 6839: 6829: 6819: 6818: 6817: 6806: 6802: 6794: 6790: 6782: 6778: 6769: 6760: 6751: 6745: 6741: 6737: 6731: 6724: 6720: 6713: 6709: 6703: 6696: 6692: 6686: 6681: 6676: 6671: 6670:branch points 6666: 6660: 6656: 6650: 6646: 6642: 6638: 6634: 6630: 6624: 6618: 6611: 6607: 6601: 6594: 6588: 6581: 6576: 6570: 6565: 6561: 6554: 6548: 6542: 6535: 6531: 6525: 6519: 6510: 6504: 6498: 6492: 6486: 6480: 6474: 6468: 6465: 6458: 6454: 6448: 6442: 6436: 6432: 6426: 6420: 6414: 6405: 6396: 6390: 6384: 6379: 6374: 6365: 6356: 6349: 6345: 6339: 6335: 6331: 6322: 6317: 6310: 6306: 6299: 6291: 6287: 6281: 6267: 6261: 6251: 6245: 6239: 6233: 6227: 6220: 6215: 6211:expansion of 6210: 6209:Taylor series 6205: 6199: 6193: 6186: 6166: 6160: 6152: 6144: 6141: 6136: 6126: 6123: 6120: 6114: 6111: 6108: 6099: 6089: 6088: 6087: 6086: 6081: 6075: 6068: 6064: 6058: 6051: 6041: 6037: 6031: 6025: 6016: 6007: 6002: 5997: 5991: 5986: 5981: 5975: 5966: 5957: 5951: 5945: 5939: 5933: 5927: 5916: 5908: 5903: 5901: 5896: 5876: 5871: 5867: 5861: 5857: 5853: 5848: 5840: 5837: 5834: 5831: 5825: 5820: 5812: 5809: 5806: 5803: 5793: 5792: 5791: 5789: 5784: 5763: 5760: 5757: 5737: 5734: 5731: 5724: 5723: 5722: 5705: 5702: 5699: 5696: 5693: 5690: 5687: 5684: 5677: 5663: 5660: 5657: 5654: 5651: 5648: 5645: 5642: 5639: 5632: 5631: 5630: 5627: 5622: 5618: 5610: 5601: 5591: 5581: 5571: 5566: 5547: 5542: 5532: 5526: 5523: 5520: 5517: 5514: 5511: 5508: 5505: 5502: 5499: 5496: 5484: 5481: 5476: 5472: 5468: 5465: 5462: 5459: 5456: 5453: 5450: 5447: 5444: 5441: 5436: 5432: 5428: 5425: 5422: 5419: 5416: 5413: 5408: 5404: 5400: 5397: 5392: 5382: 5376: 5373: 5370: 5367: 5364: 5351: 5350: 5349: 5346: 5339: 5333: 5327: 5321: 5319: 5315: 5311: 5307: 5297: 5295: 5290: 5286: 5281: 5277: 5270: 5263: 5256: 5249: 5242: 5238: 5231: 5226: 5220: 5193: 5189: 5183: 5179: 5172: 5161: 5157: 5151: 5147: 5140: 5129: 5125: 5119: 5115: 5108: 5097: 5093: 5087: 5083: 5072: 5071: 5070: 5065: 5061: 5055: 5048: 5041: 5034: 5027: 5020: 5013: 5006: 4999: 4992: 4985: 4981: 4977: 4973: 4965: 4961: 4956: 4952: 4947: 4940: 4933: 4929: 4922: 4915: 4911: 4907: 4900: 4893: 4889: 4882: 4875: 4868: 4861: 4857: 4850: 4843: 4836: 4829: 4822: 4818: 4815: 4811: 4804: 4797: 4790: 4785: 4783: 4780: 4773: 4766: 4759: 4752: 4745: 4738: 4731: 4724: 4717: 4710: 4703: 4696: 4691: 4686: 4679: 4672: 4665: 4658: 4651: 4644: 4637: 4629: 4625: 4622: 4617: 4608: 4606: 4601: 4597: 4587: 4584: 4580: 4575: 4571: 4567: 4566:limiting case 4564: 4560: 4550: 4548: 4544: 4540: 4530: 4528: 4524: 4520: 4516: 4512: 4508: 4504: 4500: 4496: 4491: 4486: 4479: 4475: 4456: 4452: 4445: 4442: 4439: 4434: 4431: 4428: 4421: 4417: 4414: 4411: 4408: 4405: 4402: 4395: 4394: 4393: 4391: 4372: 4367: 4363: 4359: 4339: 4327: 4323: 4316: 4305: 4301: 4296: 4279: 4275: 4270: 4259: 4252: 4249: 4246: 4243: 4236: 4235: 4234: 4232: 4225: 4218: 4211: 4204: 4197: 4190: 4183: 4176: 4172: 4163: 4159: 4153: 4149: 4144: 4140: 4134: 4130: 4125: 4117: 4113: 4107: 4103: 4098: 4094: 4088: 4084: 4079: 4075: 4073: 4068: 4064: 4060: 4050: 4047: 4043: 4039: 4034: 4032: 4022: 4013: 4011: 4007: 4003: 3998: 3996: 3992: 3988: 3984: 3980: 3976: 3972: 3953: 3948: 3944: 3940: 3915: 3893: 3892: 3891: 3889: 3885: 3881: 3877: 3873: 3869: 3865: 3861: 3857: 3853: 3849: 3844: 3842: 3838: 3834: 3830: 3826: 3822: 3814: 3810: 3805: 3796: 3794: 3790: 3771: 3768: 3764: 3758: 3754: 3750: 3733: 3728: 3724: 3720: 3714: 3710: 3706: 3689: 3684: 3680: 3676: 3670: 3666: 3662: 3645: 3640: 3636: 3632: 3614: 3610: 3593: 3588: 3580: 3579: 3578: 3573: 3566: 3559: 3552: 3533: 3528: 3523: 3517: 3513: 3509: 3504: 3500: 3495: 3490: 3485: 3480: 3474: 3464: 3459: 3448: 3439: 3438: 3437: 3436:of the radii 3435: 3428: 3421: 3402: 3397: 3392: 3386: 3382: 3378: 3373: 3369: 3364: 3359: 3354: 3349: 3343: 3333: 3328: 3317: 3308: 3307: 3306: 3305:in the radii 3304: 3297: 3290: 3275: 3268: 3261: 3254: 3250: 3242: 3235: 3215: 3210: 3205: 3199: 3195: 3189: 3185: 3181: 3176: 3172: 3166: 3162: 3157: 3152: 3147: 3142: 3136: 3126: 3121: 3110: 3105: 3101: 3095: 3091: 3087: 3082: 3078: 3073: 3069: 3066: 3059: 3058: 3057: 3052: 3045: 3038: 3031: 3024: 3017: 2998: 2994: 2988: 2984: 2980: 2975: 2971: 2966: 2962: 2958: 2952: 2948: 2944: 2939: 2935: 2930: 2926: 2923: 2919: 2913: 2909: 2905: 2900: 2896: 2891: 2887: 2883: 2877: 2873: 2869: 2864: 2860: 2856: 2851: 2847: 2843: 2838: 2834: 2829: 2821: 2820: 2819: 2817: 2813: 2794: 2789: 2784: 2778: 2774: 2768: 2764: 2760: 2755: 2751: 2745: 2741: 2736: 2731: 2727: 2721: 2711: 2706: 2695: 2691: 2687: 2681: 2671: 2666: 2655: 2651: 2647: 2641: 2637: 2633: 2628: 2624: 2619: 2614: 2608: 2604: 2600: 2595: 2591: 2586: 2582: 2579: 2575: 2569: 2565: 2561: 2556: 2552: 2548: 2543: 2539: 2535: 2530: 2526: 2521: 2513: 2512: 2511: 2506: 2499: 2495: 2491: 2487: 2483: 2464: 2459: 2455: 2449: 2445: 2439: 2435: 2429: 2425: 2421: 2416: 2406: 2401: 2391: 2386: 2382: 2376: 2372: 2368: 2363: 2359: 2353: 2349: 2345: 2341: 2335: 2331: 2327: 2322: 2318: 2313: 2305: 2304: 2303: 2301: 2297: 2292: 2290: 2286: 2282: 2278: 2274: 2270: 2266: 2262: 2257: 2253: 2248: 2244: 2240: 2236: 2232: 2227: 2223: 2218: 2214: 2210: 2206: 2202: 2192: 2190: 2186: 2182: 2177: 2174: 2170: 2165: 2161: 2156: 2152: 2147: 2143: 2138: 2134: 2129: 2125: 2120: 2116: 2111: 2107: 2103: 2098: 2094: 2087: 2080: 2073: 2068: 2065: 2061: 2056: 2052: 2047: 2043: 2039: 2034: 2030: 2026: 2022: 2018: 2014: 1993: 1989: 1985: 1982: 1979: 1976: 1971: 1967: 1959: 1958: 1941: 1937: 1933: 1930: 1927: 1924: 1919: 1915: 1907: 1906: 1905: 1903: 1899: 1894: 1890: 1885: 1881: 1875: 1871: 1866: 1862: 1857: 1852: 1842: 1838: 1834: 1827: 1820: 1813: 1794: 1789: 1784: 1778: 1774: 1768: 1764: 1760: 1755: 1751: 1746: 1741: 1736: 1731: 1725: 1721: 1717: 1712: 1708: 1703: 1698: 1693: 1688: 1682: 1678: 1674: 1669: 1665: 1660: 1651: 1650: 1633: 1628: 1622: 1618: 1612: 1608: 1604: 1599: 1595: 1590: 1585: 1580: 1575: 1569: 1565: 1561: 1556: 1552: 1547: 1542: 1537: 1532: 1526: 1522: 1518: 1513: 1509: 1504: 1495: 1494: 1477: 1472: 1466: 1462: 1456: 1452: 1448: 1443: 1439: 1434: 1429: 1424: 1419: 1413: 1409: 1405: 1400: 1396: 1391: 1386: 1381: 1376: 1370: 1366: 1362: 1357: 1353: 1348: 1339: 1338: 1337: 1334: 1330: 1325: 1321: 1316: 1312: 1308: 1305: 1300: 1296: 1289: 1282: 1275: 1270: 1266: 1261: 1257: 1250: 1243: 1236: 1229: 1222: 1215: 1211: 1207: 1203: 1192: 1190: 1186: 1182: 1178: 1174: 1170: 1166: 1162: 1158: 1153: 1151: 1147: 1143: 1137: 1135: 1131: 1127: 1123: 1119: 1115: 1111: 1110: 1105: 1101: 1091: 1087: 1085: 1081: 1077: 1073: 1069: 1066:As described 1059: 1057: 1053: 1049: 1045: 1041: 1037: 1033: 1029: 1025: 1021: 1017: 1006: 999: 994: 990: 988: 984: 980: 976: 972: 968: 964: 960: 956: 952: 951:trilateration 948: 944: 938: 931: 924: 920: 915: 911: 903: 896: 889: 885: 880: 876: 869: 865: 860: 856: 848: 841: 835: 831: 826: 822: 815: 811: 806: 802: 794: 787: 782: 778: 773: 769: 764: 760: 756: 749: 742: 738: 731: 724: 717: 713: 709: 700: 696: 690: 686: 679: 672: 668: 661: 653: 646: 641: 627: 625: 621: 620:Lazare Carnot 617: 613: 609: 605: 601: 596: 594: 590: 586: 582: 578: 574: 570: 566: 562: 558: 553: 551: 547: 543: 541: 528: 527:Regiomontanus 524: 519: 517: 513: 509: 505: 501: 497: 493: 489: 485: 480: 478: 474: 473: 468: 464: 460: 456: 452: 448: 438: 436: 432: 428: 424: 420: 415: 411: 405: 401: 393: 386: 379: 372: 365: 361: 354: 347: 340: 336: 329: 322: 315: 311: 304: 297: 289: 282: 275: 270: 266: 262: 257: 254: 250: 246: 241: 239: 235: 231: 227: 223: 219: 216: 212: 208: 207:tangent point 204: 199: 193: 191: 187: 176: 174: 170: 166: 165:number theory 162: 158: 154: 150: 146: 141: 139: 135: 131: 127: 123: 119: 114: 112: 108: 104: 100: 96: 92: 88: 83: 81: 76: 72: 68: 66: 50: 46: 42: 38: 29: 21: 12971:Isaac Newton 12883: 12876: 12868: 12859: 12792:Isaac Barrow 12730: (home) 12547: 12471:Newtonianism 12446:Newton scale 12409:Impact depth 12382: (1754) 12377: 12374: (1728) 12369: 12359: 12346: 12331: 12317: (1711) 12312: 12309: (1707) 12304: 12301: (1704) 12296: 12293: (1704) 12288: 12285: (1687) 12280: 12277: (1684) 12272: 12269: (1671) 12264: 12258:Publications 12201: 12188: 12030:Thomas Heath 12021: 11904: 11888:Law of sines 11744: 11676:Golden ratio 11581: 11541: 11534: 11525: 11519:(Theodosius) 11515: 11507: 11500: 11493: 11486: 11477: 11467: 11461:(Hipparchus) 11457: 11447: 11439: 11432: 11423: 11413: 11405: 11400:(Apollonius) 11396: 11388: 11376: 11351:Zeno of Elea 11111:Eratosthenes 11101:Dionysodorus 10937: 10928:. Retrieved 10912:. Retrieved 10896:. Retrieved 10893:Cut The Knot 10892: 10877: 10859:. Retrieved 10823: 10809: 10777: 10763:. Retrieved 10756:the original 10741: 10737: 10720: 10703: 10676: 10672: 10629: 10625: 10619: 10594: 10583: 10542: 10536: 10526: 10502:(3506): 62. 10499: 10493: 10483: 10449: 10445: 10434: 10430: 10406: 10402: 10383: 10374: 10370: 10359: 10355: 10347: 10343: 10335: 10331: 10328:Hachette JNP 10315: 10304: 10298: 10297:(in Latin). 10294: 10278: 10270: 10262: 10243: 10237: 10220: 10216: 10206: 10181: 10170: 10150: 10122: 10118:Mandelbrot B 10112: 10087: 10083: 10060: 10054: 10030: 10024: 10018: 9965: 9961: 9951: 9942: 9936: 9930: 9921: 9912: 9879: 9875: 9866: 9839: 9833: 9820: 9787: 9783: 9744: 9736: 9730: 9710: 9702: 9691:. Retrieved 9648: 9644: 9638: 9597:math/9909152 9590:(1): 63–66. 9587: 9583: 9574: 9549: 9545: 9539: 9514: 9510: 9504: 9479: 9475: 9443: 9439: 9429: 9420: 9416: 9410: 9388: 9383: 9378:. pp. 66–68. 9367: 9362: 9337: 9302: 9272: 9266: 9241: 9237: 9219: 9215: 9190: 9186: 9180: 9163: 9139: 9116: 9072: 9068: 9050: 9046: 9037: 9028: 9017: 9008: 8989: 8969: 8963: 8962:(in Latin). 8959: 8943: 8924: 8918: 8890: 8881: 8872: 8868: 8847: 8843: 8840:Poncelet J-V 8834: 8822: 8811: 8804: 8796: 8773: 8760: 8747: 8710: 8703: 8684: 8678: 8659: 8628: 8619: 8608: 8577: 8553: 8550:van Roomen A 8516: 8512: 8483: 8448: 8441: 8424: 8420: 8368: 8343:. Springer. 8340: 8306: 8298: 8277: 8269: 8242: 8210: 8164: 8160: 8133: 8062:Ford circles 8058:infinite set 8034:Isaac Newton 8027: 8025: 7979: 7973: 7970:Applications 7966:dimensions. 7963: 7949: 7945:hyperspheres 7940: 7938: 7930: 7917: 7915: 7898: 7894: 7882: 7871: 7867: 7861: 7838: 7807: 7798:Daniel Pedoe 7795: 7772: 7747: 7740: 7736: 7730: 7720: 7712: 7705: 7698: 7691: 7684: 7677: 7668: 7664: 7659: 7655: 7650: 7646: 7644: 7479:showed that 7465: 7448: 7444: 7433: 7411: 7404: 7400:one solution 7390: 7384: 7351: 7328: 7305: 7282: 7259: 7236: 7213: 7190: 7167: 7149:three points 7144: 7131:(in general) 7104: 7095: 7090: 7082: 7076: 7068: 7064: 7058: 7053: 7045: 7041: 7037: 7034: 7014: 6993: 6973: 6969: 6963: 6957: 6950: 6804: 6800: 6792: 6788: 6780: 6776: 6767: 6758: 6749: 6746: 6739: 6735: 6729: 6722: 6718: 6711: 6707: 6701: 6694: 6690: 6684: 6674: 6664: 6658: 6654: 6648: 6644: 6640: 6636: 6632: 6628: 6622: 6616: 6609: 6605: 6599: 6592: 6586: 6579: 6568: 6563: 6559: 6552: 6546: 6540: 6533: 6529: 6523: 6517: 6508: 6502: 6496: 6490: 6484: 6478: 6472: 6469: 6463: 6456: 6452: 6446: 6440: 6434: 6430: 6424: 6418: 6412: 6403: 6394: 6388: 6382: 6372: 6363: 6354: 6347: 6343: 6340: 6333: 6329: 6315: 6308: 6304: 6297: 6289: 6285: 6279: 6265: 6259: 6249: 6243: 6237: 6231: 6225: 6218: 6213: 6203: 6197: 6191: 6184: 6181: 6079: 6073: 6066: 6062: 6059: 6049: 6039: 6035: 6029: 6023: 6014: 6005: 5995: 5989: 5983:are conics, 5979: 5973: 5964: 5955: 5949: 5943: 5937: 5931: 5928: 5914: 5906: 5898: 5894: 5891: 5782: 5779: 5720: 5625: 5616: 5608: 5599: 5589: 5579: 5569: 5564: 5562: 5344: 5337: 5331: 5325: 5322: 5303: 5293: 5288: 5284: 5279: 5275: 5268: 5261: 5254: 5247: 5240: 5236: 5229: 5227: 5218: 5216: 5063: 5053: 5046: 5039: 5032: 5025: 5018: 5011: 5004: 4997: 4990: 4983: 4980:radical axis 4971: 4969: 4959: 4945: 4938: 4931: 4927: 4920: 4913: 4909: 4905: 4898: 4891: 4887: 4880: 4873: 4866: 4859: 4848: 4841: 4834: 4827: 4820: 4816: 4814:radical axis 4802: 4795: 4788: 4786: 4781: 4771: 4764: 4757: 4750: 4743: 4736: 4729: 4722: 4715: 4708: 4701: 4694: 4692: 4684: 4677: 4670: 4663: 4656: 4649: 4642: 4635: 4633: 4627: 4623: 4621:radical axis 4604: 4595: 4593: 4582: 4573: 4569: 4562: 4558: 4556: 4536: 4525:, forming a 4518: 4514: 4510: 4506: 4502: 4498: 4495:radical axis 4492: 4484: 4477: 4473: 4471: 4389: 4387: 4223: 4216: 4209: 4202: 4195: 4188: 4181: 4174: 4168: 4161: 4157: 4151: 4147: 4142: 4138: 4132: 4128: 4115: 4111: 4105: 4101: 4096: 4092: 4086: 4082: 4066: 4056: 4035: 4027: 3999: 3986: 3982: 3978: 3974: 3970: 3968: 3887: 3883: 3879: 3875: 3871: 3867: 3863: 3859: 3855: 3851: 3847: 3845: 3818: 3812: 3808: 3786: 3571: 3564: 3557: 3550: 3548: 3433: 3426: 3419: 3417: 3302: 3295: 3288: 3273: 3266: 3259: 3252: 3240: 3233: 3230: 3050: 3043: 3036: 3029: 3022: 3015: 3013: 2810:The product 2809: 2504: 2497: 2493: 2489: 2488:) is zero, ( 2479: 2293: 2288: 2284: 2280: 2272: 2268: 2264: 2260: 2255: 2251: 2246: 2242: 2238: 2234: 2230: 2225: 2221: 2216: 2212: 2208: 2204: 2198: 2181:real numbers 2178: 2172: 2168: 2163: 2159: 2154: 2150: 2145: 2141: 2136: 2132: 2127: 2123: 2118: 2114: 2109: 2105: 2096: 2092: 2085: 2078: 2071: 2069: 2063: 2059: 2054: 2050: 2045: 2041: 2032: 2028: 2024: 2020: 2016: 2012: 2010: 1901: 1897: 1892: 1888: 1883: 1879: 1873: 1869: 1864: 1860: 1853: 1840: 1836: 1825: 1818: 1811: 1809: 1332: 1328: 1323: 1319: 1314: 1310: 1298: 1294: 1287: 1280: 1273: 1268: 1264: 1259: 1255: 1248: 1241: 1234: 1227: 1220: 1213: 1205: 1201: 1198: 1188: 1184: 1180: 1176: 1172: 1171:case into a 1168: 1164: 1160: 1156: 1154: 1149: 1145: 1141: 1138: 1133: 1125: 1121: 1107: 1099: 1097: 1083: 1079: 1078:) or point ( 1075: 1071: 1065: 1043: 1039: 1035: 1031: 1028:eccentricity 1023: 1019: 1013: 1004: 997: 986: 982: 978: 974: 970: 966: 962: 958: 954: 947:Isaac Newton 945: 936: 929: 922: 913: 909: 901: 894: 887: 883: 878: 874: 867: 863: 858: 854: 846: 839: 833: 829: 824: 820: 813: 809: 804: 800: 792: 785: 780: 776: 771: 767: 762: 758: 747: 740: 736: 729: 722: 715: 705: 698: 694: 688: 684: 677: 670: 666: 659: 651: 644: 612:Nicolas Fuss 597: 554: 549: 545: 537: 520: 481: 470: 467:Isaac Newton 444: 427:Isaac Newton 413: 409: 403: 399: 391: 384: 377: 370: 363: 359: 352: 345: 338: 334: 327: 320: 313: 309: 302: 295: 287: 280: 273: 268: 264: 260: 258: 252: 249:tangent line 244: 242: 221: 210: 206: 202: 197: 194: 185: 182: 169:Ford circles 142: 115: 107:Isaac Newton 84: 62: 40: 34: 12871:by Paolozzi 12810:Roger Cotes 12419:Newton disc 12333:Quaestiones 12306:Arithmetica 12097:mathematics 11905:Arithmetica 11502:Ostomachion 11471:(Autolycus) 11390:Arithmetica 11166:Hippocrates 11096:Dinostratus 11081:Dicaearchus 11011:Aristarchus 10857:. Mathforum 10350:(2): 63–66. 10338:(2): 27–28. 10305:Opera Omnia 10267:de Fermat P 10223:: 136–157. 10056:Mathematika 9715:(in French) 9707:Descartes R 9423:(1): 37–47. 8970:Opera Omnia 8456:. pp. 8427:: 444–452. 8239:Greitzer SL 8235:Coxeter HSM 8206:Coolidge JL 8167:(1): 5–15. 8157:Coxeter HSM 7997:transponder 7985:World War I 7980:differences 7891:rectifiable 7735:as a poem, 7241:three lines 6990:. Because 6327:, which is 5971:. Because 3850:and radius 2812:distributes 2496:) = 0. Let 2486:square norm 2482:Lie quadric 2300:dot product 2277:orientation 1210:coordinates 1022:to a focus 585:HSM Coxeter 261:differences 80:cardinality 12988:Categories 12958:Categories 12934:XMM-Newton 12851:Depictions 12822:John Keill 12744:Apple tree 12739:Later life 12734:Early life 12314:De Analysi 12149:Babylonian 12049:arithmetic 12015:History of 11844:Apollonius 11529:(Menelaus) 11488:On Spirals 11407:Catoptrics 11346:Xenocrates 11341:Thymaridas 11326:Theodosius 11311:Theaetetus 11291:Simplicius 11281:Pythagoras 11266:Posidonius 11251:Philonides 11211:Nicomachus 11206:Metrodorus 11196:Menaechmus 11151:Hipparchus 11141:Heliodorus 11091:Diophantus 11076:Democritus 11056:Chrysippus 11026:Archimedes 11021:Apollonius 10991:Anaxagoras 10983:(timeline) 10930:2008-05-05 10914:2008-05-05 10898:2008-05-05 10861:2008-05-05 10765:2009-04-16 10589:Apostol TM 10437:: 116–126. 10033:: 99–106. 9693:2008-10-06 9580:Eppstein D 9193:: 116–126. 8887:Petersen J 8865:Gergonne J 8754:(in Latin) 8559:(in Latin) 8247:Washington 8111:References 7843:and other 7697:and radii 6678:. By the 6626:for which 4059:concentric 3862:such that 3303:difference 2070:The signs 1856:resultants 852:is either 798:is either 712:hyperbolas 593:Sophus Lie 512:Menaechmus 477:John Casey 463:hyperbolas 122:symmetries 91:hyperbolas 12773:Relations 12282:Principia 11610:Inscribed 11370:Treatises 11361:Zenodorus 11321:Theodorus 11296:Sosigenes 11241:Philolaus 11226:Oenopides 11221:Nicoteles 11216:Nicomedes 11176:Hypsicles 11071:Ctesibius 11061:Cleomedes 11046:Callippus 11031:Autolycus 11016:Aristotle 10996:Anthemius 10879:MathWorld 10794:cite book 10744:: 15–25. 10695:120042053 10634:CiteSeerX 10466:0002-9327 10423:201061558 10395:Serret JA 10377:(5): 423. 10176:Mumford D 10047:121089590 9992:0027-8424 9896:0002-9890 9812:122065577 9776:Steiner J 9688:MathWorld 9665:0002-9890 9614:0002-9890 9552:: 15–26. 9531:120984176 9496:120097802 9258:122228528 9172:252013267 9043:Cauchy AL 8966:: 95–101. 8658:(1886) . 8181:0002-9890 8082:bacterium 8029:Principia 8021:isotropic 7893:) curve ( 7880:dimension 7674:curvature 7437:inscribed 7408:inversion 7396:hyperbola 6912:: 6884:∈ 6827:Ψ 6142:: 6127:× 6121:∈ 6097:Φ 5918:tends to 5835:− 5807:− 5697:− 5688:− 5652:− 5533:∈ 5491:↔ 5398:: 5383:∈ 5292:for some 5200:¯ 5173:⋅ 5168:¯ 5136:¯ 5109:⋅ 5104:¯ 4966:(orange). 4858:point of 4539:inversion 4432:− 4418:⁡ 4409:± 4403:θ 4368:± 4360:≡ 4302:− 4250:θ 4247:⁡ 4046:Inversion 4031:inversion 3969:Thus, if 3936:¯ 3930:′ 3916:⋅ 3911:¯ 3751:∣ 3707:∣ 3663:∣ 3611:∣ 3465:− 3379:− 3334:− 3182:− 3153:− 3127:− 3088:∣ 3067:− 2981:∣ 2945:∣ 2924:− 2906:∣ 2870:− 2857:∣ 2844:− 2761:− 2732:− 2712:− 2692:⋅ 2672:− 2634:− 2601:− 2562:− 2549:∣ 2536:− 2422:− 2407:⋅ 2328:∣ 2237:), where 2102:quadratic 1761:− 1718:− 1675:− 1605:− 1562:− 1519:− 1449:− 1406:− 1363:− 1094:opposite. 1074:), line ( 1038:on which 755:hyperbola 516:parabolas 500:parabolas 479:in 1881. 475:, and by 472:Principia 198:intersect 12896:Namesake 12862:by Blake 12456:Spectrum 12397:Calculus 12366: ) 12266:Fluxions 12174:Japanese 12159:Egyptian 12102:timeline 12090:timeline 12078:timeline 12073:geometry 12066:timeline 12061:calculus 12054:timeline 12042:timeline 11745:Elements 11591:Concepts 11553:Problems 11526:Spherics 11516:Spherics 11481:(Euclid) 11427:(Euclid) 11424:Elements 11417:(Euclid) 11378:Almagest 11286:Serenus 11261:Porphyry 11201:Menelaus 11156:Hippasus 11131:Eutocius 11106:Domninus 11001:Archytas 10820:(1991). 10818:Wells, D 10786:67245614 10776:(1933). 10722:historia 10712:61042170 10667:(1973). 10665:Boyd, DW 10591:(1990). 10567:15603162 10442:Alvord B 10397:(1848). 10313:(1803). 10311:Carnot L 10301:: 17–28. 10286:(1810). 10148:(2008). 10146:Weaire D 10144:Aste T, 10120:(1983). 10104:15928775 10010:16588629 9945:: 55–71. 9920:(1803). 9918:Carnot L 9778:(1826). 9756:Archived 9739:: 91–96. 9630:14002377 9566:59444157 9334:(1990). 9222:: 14–15. 9210:(1970). 9136:(2000). 9097:30190437 9027:(1803). 9025:Carnot L 9016:(1801). 9014:Carnot L 8988:(1873). 8986:Gauss CF 8951:(1790). 8889:(1879). 8801:Simson R 8766:Boyer CB 8743:Viète F. 8625:Newton I 8607:(1687). 8605:Newton I 8574:Newton I 8552:(1596). 8393:51648067 8241:(1967). 8208:(1916). 8088:See also 8072:used on 8042:velocity 7672:are the 7078:Elements 6608:→ 6301:sending 5987:implies 5902:circles. 5900:singular 5595:−1 5587:, where 5031:; then, 4904:lies on 3890:squared 2816:bilinear 2296:bilinear 1109:Elements 567:and the 496:ellipses 253:internal 245:external 226:parallel 171:and the 161:fractals 132:such as 51:(c. 262 12414:Inertia 12402:fluxion 12298:Queries 12290:Opticks 12274:De Motu 12154:Chinese 12109:numbers 12037:algebra 11965:Related 11939:Centers 11735:Results 11605:Central 11276:Ptolemy 11271:Proclus 11236:Perseus 11191:Marinus 11171:Hypatia 11161:Hippias 11136:Geminus 11126:Eudoxus 11116:Eudemus 11086:Diocles 10547:Bibcode 10504:Bibcode 10474:2369532 10367:Dupin C 10323:, §416. 10284:Euler L 10001:1078636 9970:Bibcode 9904:2314247 9872:Pedoe D 9844:Bibcode 9826:Soddy F 9796:252–288 9792:161–184 9673:2975188 9622:2695679 9208:Pedoe D 9077:Bibcode 8949:Euler L 8803:(1734) 8656:Casey J 8533:2690380 8462:673–677 8373:Bibcode 8189:2315097 7887:regular 7876:fractal 7416:section 7134:Example 7075:in his 7050:Table 1 7036:either 6668:. The 6386:in the 6018:− 5968:− 5619:. The 5612:− 5583:− 4063:annulus 4004:on the 3014:Since ( 1845:"− + −" 1831:on the 1304:coupled 1240:) and ( 1106:in his 1052:defined 1050:can be 540:Epaphaí 469:in his 441:History 222:tangens 203:tangent 65:Epaphaí 53:BC – c. 45:tangent 12869:Newton 12860:Newton 12169:Indian 11946:Cyrene 11478:Optics 11397:Conics 11316:Theano 11306:Thales 11301:Sporus 11246:Philon 11231:Pappus 11121:Euclid 11051:Carpus 11041:Bryson 10836: 10784: 10710: 10693: 10636: 10607: 10575:626749 10573: 10565: 10495:Nature 10472: 10464: 10421: 10250: 10194: 10190:–223. 10158: 10134: 10102: 10045: 10008: 9998: 9990: 9902: 9894: 9835:Nature 9810: 9671: 9663: 9628: 9620: 9612: 9564: 9529: 9494: 9374: 9350: 9309: 9279: 9256: 9170: 9152: 9123: 9095: 8996: 8784: 8722: 8691: 8666: 8635: 8584: 8531: 8479:Pappus 8391: 8347: 8319: 8315:–141. 8286: 8257: 8187: 8179: 8078:enzyme 7870:or an 7841:sphere 7732:Nature 7645:where 7073:Euclid 7004:Ψ 6984:Ψ 6953:Ψ 6765:, and 6688:meets 6662:meets 6506:. If 6482:, say 6450:meets 6378:pencil 6325:Φ 6271:Φ 6255:Φ 6027:, and 5565:circle 5335:, and 5260:gives 5010:. Let 4756:, and 4669:, and 4415:arctan 3870:, and 3841:mirror 3839:, and 3837:rotate 3285: 3281: 2011:where 1191:case. 1104:Euclid 1048:circle 657:equal 622:, and 533:Ἐπαφαί 145:beyond 58:Ἐπαφαί 12704:table 12164:Incan 12085:logic 11861:Other 11629:Chord 11622:Axiom 11600:Angle 11256:Plato 11146:Heron 11066:Conon 10759:(PDF) 10734:(PDF) 10691:S2CID 10571:S2CID 10470:JSTOR 10419:S2CID 10388:(PDF) 10291:(PDF) 10100:S2CID 10080:(PDF) 10043:S2CID 9900:JSTOR 9808:S2CID 9749:: 51. 9669:JSTOR 9626:S2CID 9618:JSTOR 9592:arXiv 9562:S2CID 9527:S2CID 9492:S2CID 9254:S2CID 9093:S2CID 8956:(PDF) 8716:29–30 8529:JSTOR 8458:18–20 8389:S2CID 8218:–172. 8185:JSTOR 8007:(see 7993:LORAN 7816:and θ 7120:Index 7044:, or 7011:Radii 6982:, so 6814:2 = 8 6575:poles 6319:is a 6263:is a 6257:over 6047:; if 4171:Gauss 4152:inner 4143:outer 4106:inner 4097:outer 3833:shift 3028:) = ( 1114:lemma 1068:below 573:poles 435:LORAN 238:below 236:(see 215:Latin 209:or a 111:LORAN 12126:list 11414:Data 11186:Leon 11036:Bion 10834:ISBN 10800:link 10782:OCLC 10708:OCLC 10605:ISBN 10563:PMID 10462:ISSN 10407:1848 10248:ISBN 10192:ISBN 10156:ISBN 10132:ISBN 10006:PMID 9988:ISSN 9892:ISSN 9661:ISSN 9610:ISSN 9372:ISBN 9348:ISBN 9307:ISBN 9277:ISBN 9168:OCLC 9150:ISBN 9121:ISBN 8994:ISBN 8782:ISBN 8720:ISBN 8689:ISBN 8664:ISBN 8633:ISBN 8582:ISBN 8345:ISBN 8317:ISBN 8284:ISBN 8255:ISBN 8177:ISSN 8074:DVDs 8019:not 7991:and 7918:four 7889:(or 7822:dual 7711:and 7690:and 7663:and 7654:= 1/ 7439:and 7123:Code 6705:and 6590:and 6544:nor 6416:and 6410:are 6401:and 6361:and 6060:Let 5993:and 5977:and 5962:and 5947:and 5929:Let 5922:and 5750:and 5606:and 5577:and 5267:and 5253:and 5045:and 5024:and 4879:and 4856:pole 4847:and 4826:and 4801:and 4763:and 4749:and 4735:and 4707:and 4641:and 4509:and 4501:and 4483:and 4194:and 3878:and 3570:and 3425:and 3294:and 3258:and 2503:and 2480:The 2084:and 2058:and 2023:and 1896:and 1824:and 1327:and 1309:for 1293:and 1226:), ( 985:and 965:and 928:and 919:foci 775:and 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