2870:
2922:
473:
1819:
132:
53:
4212:(broadly defined as a figure with three vertices connected by curves that are concave to the exterior of the deltoid, making the interior points a non-convex set). The vertices of the deltoid are at the midpoints of the medians; all points inside the deltoid are on three different area bisectors, while all points outside it are on just one.
2355:
2854:
3657:
lies at the intersection of six angle bisectors. These are the internal angle bisectors at two opposite vertex angles, the external angle bisectors (supplementary angle bisectors) at the other two vertex angles, and the external angle bisectors at the angles formed where the
2510:, one draws a circle whose center is the vertex. The circle meets the angle at two points: one on each leg. Using each of these points as a center, draw two circles of the same size. The intersection of the circles (two points) determines a line that is the angle bisector.
4096:
are the line segments that connect the midpoints of opposite sides, hence each bisecting two sides. The two bimedians and the line segment joining the midpoints of the diagonals are concurrent at a point called the "vertex centroid" and are all bisected by this point.
3585:
4409:
A plane that divides two opposite edges of a tetrahedron in a given ratio also divides the volume of the tetrahedron in the same ratio. Thus any plane containing a bimedian (connector of opposite edges' midpoints) of a tetrahedron bisects the volume of the tetrahedron
1484:
1982:
1208:
4165:; indeed, they are the only area bisectors that go through the centroid. Three other area bisectors are parallel to the triangle's sides; each of these intersects the other two sides so as to divide them into segments with the proportions
3741:
bisector of a side of a triangle is the segment, falling entirely on and inside the triangle, of the line that perpendicularly bisects that side. The three perpendicular bisectors of a triangle's three sides intersect at the
3722:
if it has uniform density; thus any line through a triangle's centroid and one of its vertices bisects the opposite side. The centroid is twice as close to the midpoint of any one side as it is to the opposite vertex.
2110:
455:
1037:
2667:
897:
4290:
2100:
2907:
Three intersection points, two of them between an interior angle bisector and the opposite side, and the third between the other exterior angle bisector and the opposite side extended, are collinear.
4021:
3933:
3846:
1637:
3757:
the two shortest sides' perpendicular bisectors (extended beyond their opposite triangle sides to the circumcenter) are divided by their respective intersecting triangle sides in equal proportions.
3380:
3189:
1783:
1710:
3121:
1300:
2473:
1552:
563:
4100:
The four "maltitudes" of a convex quadrilateral are the perpendiculars to a side through the midpoint of the opposite side, hence bisecting the latter side. If the quadrilateral is
2659:
1310:
2591:
941:
795:
4337:). There are either one, two, or three of these for any given triangle. A line through the incenter bisects one of the area or perimeter if and only if it also bisects the other.
1856:
1082:
1072:
4201:. These six lines are concurrent three at a time: in addition to the three medians being concurrent, any one median is concurrent with two of the side-parallel area bisectors.
4199:
3714:
and the midpoint of the opposite side, so it bisects that side (though not in general perpendicularly). The three medians intersect each other at a point which is called the
3280:
255:
4053:
3345:
3030:
2462:
4223:
to the extended sides of the triangle. The ratio of the area of the envelope of area bisectors to the area of the triangle is invariant for all triangles, and equals
2937:'s side is divided into by a line that bisects the opposite angle. It equates their relative lengths to the relative lengths of the other two sides of the triangle.
3372:
3219:
2975:
565:, whose centers are the endpoints of the segment. The line determined by the points of intersection of the two circles is the perpendicular bisector of the segment.
821:
720:
4076:
2389:
743:
512:
172:
4326:
of a triangle is a line segment having one endpoint at one of the three vertices of the triangle and bisecting the perimeter. The three splitters concur at the
3050:
2409:
694:
674:
654:
631:
605:
585:
192:
4129:
274:
3746:(the center of the circle through the three vertices). Thus any line through a triangle's circumcenter and perpendicular to a side bisects that side.
2499:, or line segment that divides an angle of less than 180° into two equal angles. The 'exterior' or 'external bisector' is the line that divides the
4333:
Any line through a triangle that splits both the triangle's area and its perimeter in half goes through the triangle's incenter (the center of its
2350:{\displaystyle \quad (a_{1}-b_{1})x+(a_{2}-b_{2})y+(a_{3}-b_{3})z={\tfrac {1}{2}}(a_{1}^{2}-b_{1}^{2}+a_{2}^{2}-b_{2}^{2}+a_{3}^{2}-b_{3}^{2})\;.}
2503:(of 180° minus the original angle), formed by one side forming the original angle and the extension of the other side, into two equal angles.
286:
56:
Line DE bisects line AB at D, line EF is a perpendicular bisector of segment AD at C, and line EF is the interior bisector of right angle AED
950:
2849:{\displaystyle {\frac {l_{1}x+m_{1}y+n_{1}}{\sqrt {l_{1}^{2}+m_{1}^{2}}}}=\pm {\frac {l_{2}x+m_{2}y+n_{2}}{\sqrt {l_{2}^{2}+m_{2}^{2}}}}.}
826:
3599:
4399:
4226:
1989:
3580:{\displaystyle {\frac {(b+c)^{2}}{bc}}t_{a}^{2}+{\frac {(c+a)^{2}}{ca}}t_{b}^{2}+{\frac {(a+b)^{2}}{ab}}t_{c}^{2}=(a+b+c)^{2}.}
3938:
3850:
3763:
1559:
481:
3132:
1715:
1642:
3058:
4125:), then the perpendicular to a side from the point of intersection of the diagonals always bisects the opposite side.
2492:
measures. An angle only has one bisector. Each point of an angle bisector is equidistant from the sides of the angle.
1215:
3753:
the circumcenter divides the interior perpendicular bisectors of the two shortest sides in equal proportions. In an
2517:(dividing it into three equal parts) cannot be achieved with the compass and ruler alone (this was first proved by
2513:
The proof of the correctness of this construction is fairly intuitive, relying on the symmetry of the problem. The
4394:
If a line segment connecting the diagonals of a quadrilateral bisects both diagonals, then this line segment (the
17:
4471:
Oxman, Victor. "On the existence of triangles with given lengths of one side and two adjacent angle bisectors",
3684:
at any point bisects the angle between the line joining the point to the focus and the line from the point and
1479:{\displaystyle \quad (a_{1}-b_{1})x+(a_{2}-b_{2})y={\tfrac {1}{2}}(a_{1}^{2}-b_{1}^{2}+a_{2}^{2}-b_{2}^{2})\;.}
1495:
517:
4502:
1977:{\displaystyle \quad {\vec {x}}\cdot ({\vec {a}}-{\vec {b}})={\tfrac {1}{2}}({\vec {a}}^{2}-{\vec {b}}^{2}).}
1203:{\displaystyle \quad {\vec {x}}\cdot ({\vec {a}}-{\vec {b}})={\tfrac {1}{2}}({\vec {a}}^{2}-{\vec {b}}^{2}).}
2596:
4738:
4516:
Oxman, Victor, "A purely geometric proof of the uniqueness of a triangle with prescribed angle bisectors",
2531:
902:
756:
4500:
Mironescu, P., and
Panaitopol, L., "The existence of a triangle with prescribed angle bisector lengths",
4115:
3654:
1042:
4168:
3293:
are the side lengths opposite vertices B and C; and the side opposite A is divided in the proportion
3235:
207:
4303:
of the triangle and has one endpoint at the midpoint of one of the three sides. The three cleavers
4111:
4026:
3711:
3630:
2507:
3307:
2980:
2414:
567:
Because the construction of the bisector is done without the knowledge of the segment's midpoint
105:
46:
4205:
4157:
of the triangle (which connect the sides' midpoints with the opposite vertices), and these are
3715:
2916:
2489:
277:
4132:
forms a quadrilateral from the perpendicular bisectors of the sides of another quadrilateral.
3671:
69:
3590:
No two non-congruent triangles share the same set of three internal angle bisector lengths.
4618:
4101:
3618:
3350:
3197:
2948:
38:
4425:
8:
4323:
2500:
1798:
800:
699:
4680:
4058:
2371:
725:
494:
154:
4722:
4659:
4599:
4550:
4365:
through the center bisect the area and perimeter. In the case of a circle they are the
4357:
All area bisectors and perimeter bisectors of a circle or other ellipse go through the
4296:
4086:
3035:
2394:
679:
659:
639:
616:
590:
570:
177:
73:
4515:
4476:
263:
4698:
4426:
Weisstein, Eric W. "Exterior Angle
Bisector." From MathWorld--A Wolfram Web Resource.
4358:
4154:
3707:
4686:
4591:
4362:
4304:
4158:
4105:
3626:
3622:
2874:
2514:
1793:
Perpendicular line segment bisectors were used solving various geometric problems:
109:
42:
4701:
4690:
4531:
4316:
3754:
1811:
97:
4668:
4632:
2921:
2896:
Three intersection points, each of an external angle bisector with the opposite
4674:
4308:
4090:
3750:
3719:
3611:
2890:
2886:
2518:
2496:
485:
4732:
4346:
4209:
4119:
4093:
3738:
3685:
3659:
3614:
2897:
2525:
1833:
944:
141:
4530:
Weisstein, Eric W. "Quadrilateral." From MathWorld--A Wolfram Web
Resource.
4663:
3743:
3732:
2901:
89:
4631:
Weisstein, Eric W. "Tetrahedron." From MathWorld--A Wolfram Web
Resource.
4545:
Mitchell, Douglas W. (2013), "Perpendicular
Bisectors of Triangle Sides",
3221:
and if this bisector divides the side opposite A into segments of lengths
472:
4395:
4327:
2472:
947:
of the perpendicular line segment bisector. Hence its vector equation is
195:
2661:
then the internal and external bisectors are given by the two equations
450:{\displaystyle |XA|^{2}=|XM|^{2}+|MA|^{2}=|XM|^{2}+|MB|^{2}=|XB|^{2}\;.}
72:
parts (having the same shape and size). Usually it involves a bisecting
4718:
4603:
4220:
4213:
3621:(that is, the four intersection points of adjacent angle bisectors are
722:, and the perpendicular to be constructed is the one bisecting segment
31:
1818:
1032:{\displaystyle ({\vec {x}}-{\vec {m}})\cdot ({\vec {a}}-{\vec {b}})=0}
4706:
4300:
4216:
4122:
4595:
144:
bisector of a line segment is a line which meets the segment at its
4383:
4366:
4334:
4312:
4162:
4150:
3760:
For any triangle the interior perpendicular bisectors are given by
3681:
2934:
2878:
1841:
1805:
145:
131:
85:
61:
2528:. If the angle is formed by the two lines given algebraically as
464:
is usually used for the construction of a perpendicular bisector:
4582:
Dunn, Jas. A.; Pretty, Jas. E. (May 1972). "Halving a triangle".
3677:
3642:
1848:
Its vector equation is literally the same as in the plane case:
1826:
892:{\displaystyle M:{\vec {m}}={\tfrac {{\vec {a}}+{\vec {b}}}{2}}}
3194:
If the internal bisector of angle A in triangle ABC has length
2930:
2495:
The 'interior' or 'internal bisector' of an angle is the line,
3304:
If the internal bisectors of angles A, B, and C have lengths
2485:
101:
4285:{\displaystyle {\tfrac {3}{4}}\log _{e}(2)-{\tfrac {1}{2}},}
514:
is bisected by drawing intersecting circles of equal radius
467:
52:
4146:
2869:
2095:{\displaystyle A=(a_{1},a_{2},a_{3}),B=(b_{1},b_{2},b_{3})}
4717:
This article incorporates material from Angle bisector on
2929:
The angle bisector theorem is concerned with the relative
607:
as the intersection of the bisector and the line segment.
4135:
4108:
at (all meet at) a common point called the "anticenter".
3052:, then the length of the internal bisector of angle A is
4016:{\displaystyle p_{c}={\tfrac {2cT}{a^{2}-b^{2}+c^{2}}},}
3928:{\displaystyle p_{b}={\tfrac {2bT}{a^{2}+b^{2}-c^{2}}},}
3841:{\displaystyle p_{a}={\tfrac {2aT}{a^{2}+b^{2}-c^{2}}},}
1632:{\displaystyle \;m=-{\tfrac {b_{1}-a_{1}}{b_{2}-a_{2}}}}
1074:
and expanding the equation leads to the vector equation
1814:
boundaries consist of segments of such lines or planes.
266:
80:. The most often considered types of bisectors are the
27:
Division of something into two equal or congruent parts
4696:
4268:
4231:
3956:
3868:
3781:
3691:
3184:{\displaystyle {\frac {2bc}{b+c}}\cos {\frac {A}{2}}.}
2476:
Bisection of an angle using a compass and straightedge
2221:
1913:
1778:{\displaystyle \;y_{0}={\tfrac {1}{2}}(a_{2}+b_{2})\;}
1734:
1705:{\displaystyle \;x_{0}={\tfrac {1}{2}}(a_{1}+b_{1})\;}
1661:
1574:
1386:
1139:
852:
610:
This construction is in fact used when constructing a
528:
119:
4229:
4171:
4061:
4029:
3941:
3853:
3766:
3383:
3353:
3310:
3238:
3200:
3135:
3061:
3038:
2983:
2951:
2670:
2599:
2534:
2417:
2397:
2374:
2113:
1992:
1859:
1718:
1645:
1562:
1498:
1313:
1218:
1085:
1045:
953:
905:
829:
803:
759:
728:
702:
682:
662:
642:
619:
593:
573:
520:
497:
289:
210:
180:
157:
4551:
http://forumgeom.fau.edu/FG2013volume13/FG201307.pdf
4319:. The cleavers are parallel to the angle bisectors.
2524:
The internal and external bisectors of an angle are
4477:
http://forumgeom.fau.edu/FG2004volume4/FG200425.pdf
3116:{\displaystyle {\frac {2{\sqrt {bcs(s-a)}}}{b+c}},}
2864:
484:, whose possibility depends on the ability to draw
4284:
4193:
4070:
4047:
4015:
3927:
3840:
3710:of a triangle is a line segment going through one
3579:
3366:
3339:
3274:
3213:
3183:
3115:
3044:
3024:
2969:
2848:
2653:
2585:
2456:
2403:
2383:
2349:
2094:
1976:
1777:
1704:
1631:
1546:
1478:
1294:
1202:
1066:
1031:
935:
891:
815:
789:
737:
714:
688:
668:
648:
625:
599:
579:
557:
506:
449:
268:
249:
186:
166:
4643:Altshiller-Court, N. "The tetrahedron." Ch. 4 in
4299:of a triangle is a line segment that bisects the
480:In classical geometry, the bisection is a simple
37:For the bisection theorem in measure theory, see
4730:
4723:Creative Commons Attribution/Share-Alike License
4555:
4145:There is an infinitude of lines that bisect the
3600:integer triangles with a rational angle bisector
4669:Angle Bisector definition. Math Open Reference
4532:http://mathworld.wolfram.com/Quadrilateral.html
2873:The interior angle bisectors of a triangle are
1295:{\displaystyle A=(a_{1},a_{2}),B=(b_{1},b_{2})}
68:is the division of something into two equal or
4675:Line Bisector definition. Math Open Reference
4616:Kodokostas, Dimitrios, "Triangle Equalizers,"
4104:(inscribed in a circle), these maltitudes are
3648:
2368:The perpendicular bisector plane of a segment
174:also has the property that each of its points
4633:http://mathworld.wolfram.com/Tetrahedron.html
1827:Perpendicular line segment bisectors in space
151:The perpendicular bisector of a line segment
4687:Animated instructions for bisecting an angle
3629:. In the latter case the quadrilateral is a
2363:(see above) is literally true in space, too:
104:(that divides it into two equal angles). In
587:, the construction is used for determining
108:, bisection is usually done by a bisecting
4581:
4541:
4539:
3726:
3672:Parabola § Tangent bisection property
2418:
2343:
2102:one gets the equation in coordinate form:
1774:
1719:
1701:
1646:
1563:
1472:
1302:one gets the equation in coordinate form:
443:
4454:
4452:
4450:
4448:
4446:
4372:
4208:of the infinitude of area bisectors is a
4114:states that if a cyclic quadrilateral is
2910:
476:Construction by straight edge and compass
468:Construction by straight edge and compass
4440:, Dover Publications, 2007 (orig. 1957).
2920:
2881:of the triangle, as seen in the diagram.
2868:
2471:
1817:
1547:{\displaystyle \quad y=m(x-x_{0})+y_{0}}
558:{\displaystyle r>{\tfrac {1}{2}}|AB|}
471:
135:Perpendicular bisector of a line segment
130:
51:
4577:
4575:
4573:
4571:
4536:
2904:(fall on the same line as each other).
797:are the position vectors of two points
14:
4731:
4443:
4386:of a parallelogram bisect each other.
4136:Area bisectors and perimeter bisectors
2945:If the side lengths of a triangle are
2654:{\displaystyle l_{2}x+m_{2}y+n_{2}=0,}
488:of equal radii and different centers:
4697:
4352:
4215:The sides of the deltoid are arcs of
2586:{\displaystyle l_{1}x+m_{1}y+n_{1}=0}
936:{\displaystyle {\vec {a}}-{\vec {b}}}
790:{\displaystyle {\vec {a}},{\vec {b}}}
482:compass and straightedge construction
4568:
4349:bisects the area and the perimeter.
3593:
4404:
4345:Any line through the midpoint of a
4130:perpendicular bisector construction
3692:Bisectors of the sides of a polygon
656:: drawing a circle whose center is
120:Perpendicular line segment bisector
41:. For the root-finding method, see
24:
4389:
4377:
4292:i.e. 0.019860... or less than 2%.
4080:
3610:The internal angle bisectors of a
1804:Construction of the center of the
1067:{\displaystyle {\vec {m}}=\cdots }
612:line perpendicular to a given line
25:
4750:
4653:
4462:, Dover Publ., 2007 (orig. 1929).
3032:and A is the angle opposite side
2467:
1840:, which meets the segment at its
676:such that it intersects the line
96:, a line that passes through the
84:, a line that passes through the
4693:Using a compass and straightedge
4340:
3605:
2865:Concurrencies and collinearities
1836:bisector of a line segment is a
1797:Construction of the center of a
4637:
4625:
4610:
4194:{\displaystyle {\sqrt {2}}+1:1}
3275:{\displaystyle t_{a}^{2}+mn=bc}
2925:In this diagram, BD:DC = AB:AC.
2114:
1860:
1788:
1499:
1314:
1086:
250:{\displaystyle \quad |XA|=|XB|}
211:
4721:, which is licensed under the
4524:
4509:
4494:
4481:
4465:
4430:
4419:
4261:
4255:
3718:of the triangle, which is its
3565:
3546:
3508:
3495:
3454:
3441:
3400:
3387:
3091:
3079:
3008:
2990:
2889:and the bisector of the other
2450:
2439:
2431:
2420:
2340:
2232:
2211:
2185:
2176:
2150:
2141:
2115:
2089:
2050:
2038:
1999:
1968:
1956:
1934:
1924:
1906:
1900:
1885:
1876:
1867:
1771:
1745:
1698:
1672:
1528:
1509:
1469:
1397:
1376:
1350:
1341:
1315:
1289:
1263:
1251:
1225:
1194:
1182:
1160:
1150:
1132:
1126:
1111:
1102:
1093:
1052:
1020:
1014:
999:
990:
984:
978:
963:
954:
927:
912:
876:
861:
842:
781:
766:
551:
540:
433:
421:
407:
395:
381:
369:
355:
343:
329:
317:
303:
291:
243:
232:
224:
213:
13:
1:
4503:American Mathematical Monthly
4413:
4140:
4048:{\displaystyle a\geq b\geq c}
3696:
126:
4681:Perpendicular Line Bisector.
4622:83, April 2010, pp. 141-146.
4398:) is itself bisected by the
4309:center of the Spieker circle
3660:extensions of opposite sides
3340:{\displaystyle t_{a},t_{b},}
3025:{\displaystyle s=(a+b+c)/2,}
748:
198:from segment AB's endpoints:
7:
4460:Advanced Euclidean Geometry
3665:
3655:ex-tangential quadrilateral
3649:Ex-tangential quadrilateral
3126:or in trigonometric terms,
2933:of the two segments that a
2859:
2457:{\displaystyle \;|XA|=|XB|}
10:
4755:
4645:Modern Pure Solid Geometry
4561:Altshiller-Court, Nathan,
4307:at (all pass through) the
3730:
3701:
3669:
3636:
2940:
2914:
36:
29:
3645:bisects opposite angles.
4584:The Mathematical Gazette
4491:93, March 2009, 115-116.
4153:. Three of them are the
3631:tangential quadrilateral
2508:straightedge and compass
2506:To bisect an angle with
30:Not to be confused with
4683:With interactive applet
4677:With interactive applet
4671:With interactive applet
3727:Perpendicular bisectors
823:, then its midpoint is
260:The proof follows from
106:three-dimensional space
47:Bisect (disambiguation)
4373:Bisectors of diagonals
4286:
4195:
4072:
4049:
4017:
3929:
3842:
3581:
3368:
3341:
3276:
3215:
3185:
3117:
3046:
3026:
2971:
2926:
2917:Angle bisector theorem
2911:Angle bisector theorem
2882:
2877:in a point called the
2850:
2655:
2587:
2515:trisection of an angle
2477:
2458:
2405:
2385:
2351:
2096:
1978:
1823:
1779:
1706:
1633:
1548:
1480:
1296:
1204:
1068:
1033:
937:
893:
817:
791:
739:
716:
690:
670:
650:
627:
601:
581:
559:
508:
477:
451:
270:
251:
188:
168:
136:
57:
45:. For other uses, see
4287:
4196:
4112:Brahmagupta's theorem
4073:
4050:
4018:
3930:
3843:
3582:
3369:
3367:{\displaystyle t_{c}}
3342:
3277:
3216:
3214:{\displaystyle t_{a}}
3186:
3118:
3047:
3027:
2972:
2970:{\displaystyle a,b,c}
2924:
2885:The bisectors of two
2872:
2851:
2656:
2588:
2488:into two angles with
2475:
2459:
2406:
2386:
2352:
2097:
1979:
1821:
1780:
1707:
1634:
1549:
1481:
1297:
1205:
1069:
1034:
938:
894:
818:
792:
740:
717:
691:
671:
651:
628:
602:
582:
560:
509:
475:
452:
271:
252:
189:
169:
134:
55:
4619:Mathematics Magazine
4565:, Dover Publ., 2007.
4489:Mathematical Gazette
4227:
4169:
4059:
4027:
4023:where the sides are
3939:
3851:
3764:
3619:cyclic quadrilateral
3381:
3351:
3308:
3236:
3198:
3133:
3059:
3036:
2981:
2977:, the semiperimeter
2949:
2668:
2597:
2532:
2415:
2395:
2372:
2111:
1990:
1857:
1716:
1643:
1560:
1496:
1311:
1216:
1083:
1043:
951:
903:
827:
801:
757:
726:
700:
680:
660:
640:
617:
591:
571:
518:
495:
287:
264:
208:
178:
155:
39:Ham sandwich theorem
4739:Elementary geometry
4547:Forum Geometricorum
4518:Forum Geometricorum
4473:Forum Geometricorum
4458:Johnson, Roger A.,
3653:The excenter of an
3641:Each diagonal of a
3542:
3488:
3434:
3253:
2839:
2821:
2750:
2732:
2501:supplementary angle
2339:
2321:
2303:
2285:
2267:
2249:
1468:
1450:
1432:
1414:
816:{\displaystyle A,B}
715:{\displaystyle A,B}
278:Pythagoras' theorem
4699:Weisstein, Eric W.
4660:The Angle Bisector
4520:8 (2008): 197–200.
4506:101 (1994): 58–60.
4475:4, 2004, 215–218.
4353:Circle and ellipse
4282:
4277:
4240:
4191:
4161:at the triangle's
4071:{\displaystyle T.}
4068:
4045:
4013:
4008:
3925:
3920:
3838:
3833:
3706:Each of the three
3688:to the directrix.
3577:
3528:
3474:
3420:
3364:
3337:
3272:
3239:
3211:
3181:
3113:
3042:
3022:
2967:
2927:
2883:
2846:
2825:
2807:
2736:
2718:
2651:
2583:
2478:
2454:
2401:
2391:has for any point
2384:{\displaystyle AB}
2381:
2347:
2325:
2307:
2289:
2271:
2253:
2235:
2230:
2092:
1974:
1922:
1824:
1775:
1743:
1702:
1670:
1629:
1627:
1544:
1476:
1454:
1436:
1418:
1400:
1395:
1292:
1200:
1148:
1064:
1029:
933:
889:
887:
813:
787:
738:{\displaystyle AB}
735:
712:
686:
666:
646:
623:
597:
577:
555:
537:
507:{\displaystyle AB}
504:
478:
447:
247:
184:
167:{\displaystyle AB}
164:
137:
112:, also called the
58:
4487:Simons, Stuart.
4438:Analytical Conics
4330:of the triangle.
4276:
4239:
4177:
4007:
3919:
3832:
3594:Integer triangles
3526:
3472:
3418:
3176:
3160:
3108:
3094:
3045:{\displaystyle a}
2841:
2840:
2752:
2751:
2404:{\displaystyle X}
2229:
1959:
1937:
1921:
1903:
1888:
1870:
1742:
1669:
1626:
1394:
1185:
1163:
1147:
1129:
1114:
1096:
1055:
1017:
1002:
981:
966:
930:
915:
886:
879:
864:
845:
784:
769:
689:{\displaystyle g}
669:{\displaystyle P}
649:{\displaystyle P}
626:{\displaystyle g}
600:{\displaystyle M}
580:{\displaystyle M}
536:
187:{\displaystyle X}
16:(Redirected from
4746:
4712:
4711:
4691:bisecting a line
4648:
4647:: Chelsea, 1979.
4641:
4635:
4629:
4623:
4614:
4608:
4607:
4590:(396): 105–108.
4579:
4566:
4563:College Geometry
4559:
4553:
4543:
4534:
4528:
4522:
4513:
4507:
4498:
4492:
4485:
4479:
4469:
4463:
4456:
4441:
4434:
4428:
4423:
4405:Volume bisectors
4400:vertex centroid.
4291:
4289:
4288:
4283:
4278:
4269:
4251:
4250:
4241:
4232:
4200:
4198:
4197:
4192:
4178:
4173:
4077:
4075:
4074:
4069:
4055:and the area is
4054:
4052:
4051:
4046:
4022:
4020:
4019:
4014:
4009:
4006:
4005:
4004:
3992:
3991:
3979:
3978:
3968:
3957:
3951:
3950:
3934:
3932:
3931:
3926:
3921:
3918:
3917:
3916:
3904:
3903:
3891:
3890:
3880:
3869:
3863:
3862:
3847:
3845:
3844:
3839:
3834:
3831:
3830:
3829:
3817:
3816:
3804:
3803:
3793:
3782:
3776:
3775:
3625:), or they are
3586:
3584:
3583:
3578:
3573:
3572:
3541:
3536:
3527:
3525:
3517:
3516:
3515:
3493:
3487:
3482:
3473:
3471:
3463:
3462:
3461:
3439:
3433:
3428:
3419:
3417:
3409:
3408:
3407:
3385:
3373:
3371:
3370:
3365:
3363:
3362:
3346:
3344:
3343:
3338:
3333:
3332:
3320:
3319:
3281:
3279:
3278:
3273:
3252:
3247:
3220:
3218:
3217:
3212:
3210:
3209:
3190:
3188:
3187:
3182:
3177:
3169:
3161:
3159:
3148:
3137:
3122:
3120:
3119:
3114:
3109:
3107:
3096:
3095:
3069:
3063:
3051:
3049:
3048:
3043:
3031:
3029:
3028:
3023:
3015:
2976:
2974:
2973:
2968:
2893:are concurrent.
2855:
2853:
2852:
2847:
2842:
2838:
2833:
2820:
2815:
2806:
2805:
2804:
2803:
2788:
2787:
2772:
2771:
2761:
2753:
2749:
2744:
2731:
2726:
2717:
2716:
2715:
2714:
2699:
2698:
2683:
2682:
2672:
2660:
2658:
2657:
2652:
2641:
2640:
2625:
2624:
2609:
2608:
2592:
2590:
2589:
2584:
2576:
2575:
2560:
2559:
2544:
2543:
2463:
2461:
2460:
2455:
2453:
2442:
2434:
2423:
2410:
2408:
2407:
2402:
2390:
2388:
2387:
2382:
2356:
2354:
2353:
2348:
2338:
2333:
2320:
2315:
2302:
2297:
2284:
2279:
2266:
2261:
2248:
2243:
2231:
2222:
2210:
2209:
2197:
2196:
2175:
2174:
2162:
2161:
2140:
2139:
2127:
2126:
2101:
2099:
2098:
2093:
2088:
2087:
2075:
2074:
2062:
2061:
2037:
2036:
2024:
2023:
2011:
2010:
1983:
1981:
1980:
1975:
1967:
1966:
1961:
1960:
1952:
1945:
1944:
1939:
1938:
1930:
1923:
1914:
1905:
1904:
1896:
1890:
1889:
1881:
1872:
1871:
1863:
1844:perpendicularly.
1784:
1782:
1781:
1776:
1770:
1769:
1757:
1756:
1744:
1735:
1729:
1728:
1711:
1709:
1708:
1703:
1697:
1696:
1684:
1683:
1671:
1662:
1656:
1655:
1638:
1636:
1635:
1630:
1628:
1625:
1624:
1623:
1611:
1610:
1600:
1599:
1598:
1586:
1585:
1575:
1553:
1551:
1550:
1545:
1543:
1542:
1527:
1526:
1485:
1483:
1482:
1477:
1467:
1462:
1449:
1444:
1431:
1426:
1413:
1408:
1396:
1387:
1375:
1374:
1362:
1361:
1340:
1339:
1327:
1326:
1301:
1299:
1298:
1293:
1288:
1287:
1275:
1274:
1250:
1249:
1237:
1236:
1209:
1207:
1206:
1201:
1193:
1192:
1187:
1186:
1178:
1171:
1170:
1165:
1164:
1156:
1149:
1140:
1131:
1130:
1122:
1116:
1115:
1107:
1098:
1097:
1089:
1073:
1071:
1070:
1065:
1057:
1056:
1048:
1038:
1036:
1035:
1030:
1019:
1018:
1010:
1004:
1003:
995:
983:
982:
974:
968:
967:
959:
942:
940:
939:
934:
932:
931:
923:
917:
916:
908:
898:
896:
895:
890:
888:
882:
881:
880:
872:
866:
865:
857:
853:
847:
846:
838:
822:
820:
819:
814:
796:
794:
793:
788:
786:
785:
777:
771:
770:
762:
744:
742:
741:
736:
721:
719:
718:
713:
695:
693:
692:
687:
675:
673:
672:
667:
655:
653:
652:
647:
632:
630:
629:
624:
606:
604:
603:
598:
586:
584:
583:
578:
564:
562:
561:
556:
554:
543:
538:
529:
513:
511:
510:
505:
456:
454:
453:
448:
442:
441:
436:
424:
416:
415:
410:
398:
390:
389:
384:
372:
364:
363:
358:
346:
338:
337:
332:
320:
312:
311:
306:
294:
275:
273:
272:
269:{\displaystyle }
267:
256:
254:
253:
248:
246:
235:
227:
216:
193:
191:
190:
185:
173:
171:
170:
165:
148:perpendicularly.
82:segment bisector
76:, also called a
43:Bisection method
21:
4754:
4753:
4749:
4748:
4747:
4745:
4744:
4743:
4729:
4728:
4702:"Line Bisector"
4656:
4651:
4642:
4638:
4630:
4626:
4615:
4611:
4596:10.2307/3615256
4580:
4569:
4560:
4556:
4544:
4537:
4529:
4525:
4514:
4510:
4499:
4495:
4486:
4482:
4470:
4466:
4457:
4444:
4435:
4431:
4424:
4420:
4416:
4407:
4392:
4380:
4375:
4369:of the circle.
4355:
4343:
4317:medial triangle
4311:, which is the
4267:
4246:
4242:
4230:
4228:
4225:
4224:
4172:
4170:
4167:
4166:
4143:
4138:
4083:
4060:
4057:
4056:
4028:
4025:
4024:
4000:
3996:
3987:
3983:
3974:
3970:
3969:
3958:
3955:
3946:
3942:
3940:
3937:
3936:
3912:
3908:
3899:
3895:
3886:
3882:
3881:
3870:
3867:
3858:
3854:
3852:
3849:
3848:
3825:
3821:
3812:
3808:
3799:
3795:
3794:
3783:
3780:
3771:
3767:
3765:
3762:
3761:
3755:obtuse triangle
3735:
3729:
3704:
3699:
3694:
3674:
3668:
3651:
3639:
3608:
3596:
3568:
3564:
3537:
3532:
3518:
3511:
3507:
3494:
3492:
3483:
3478:
3464:
3457:
3453:
3440:
3438:
3429:
3424:
3410:
3403:
3399:
3386:
3384:
3382:
3379:
3378:
3358:
3354:
3352:
3349:
3348:
3328:
3324:
3315:
3311:
3309:
3306:
3305:
3248:
3243:
3237:
3234:
3233:
3205:
3201:
3199:
3196:
3195:
3168:
3149:
3138:
3136:
3134:
3131:
3130:
3097:
3068:
3064:
3062:
3060:
3057:
3056:
3037:
3034:
3033:
3011:
2982:
2979:
2978:
2950:
2947:
2946:
2943:
2919:
2913:
2887:exterior angles
2867:
2862:
2834:
2829:
2816:
2811:
2799:
2795:
2783:
2779:
2767:
2763:
2762:
2760:
2745:
2740:
2727:
2722:
2710:
2706:
2694:
2690:
2678:
2674:
2673:
2671:
2669:
2666:
2665:
2636:
2632:
2620:
2616:
2604:
2600:
2598:
2595:
2594:
2571:
2567:
2555:
2551:
2539:
2535:
2533:
2530:
2529:
2470:
2449:
2438:
2430:
2419:
2416:
2413:
2412:
2396:
2393:
2392:
2373:
2370:
2369:
2364:
2334:
2329:
2316:
2311:
2298:
2293:
2280:
2275:
2262:
2257:
2244:
2239:
2220:
2205:
2201:
2192:
2188:
2170:
2166:
2157:
2153:
2135:
2131:
2122:
2118:
2112:
2109:
2108:
2083:
2079:
2070:
2066:
2057:
2053:
2032:
2028:
2019:
2015:
2006:
2002:
1991:
1988:
1987:
1962:
1951:
1950:
1949:
1940:
1929:
1928:
1927:
1912:
1895:
1894:
1880:
1879:
1862:
1861:
1858:
1855:
1854:
1829:
1812:Voronoi diagram
1791:
1765:
1761:
1752:
1748:
1733:
1724:
1720:
1717:
1714:
1713:
1692:
1688:
1679:
1675:
1660:
1651:
1647:
1644:
1641:
1640:
1619:
1615:
1606:
1602:
1601:
1594:
1590:
1581:
1577:
1576:
1573:
1561:
1558:
1557:
1555:
1538:
1534:
1522:
1518:
1497:
1494:
1493:
1489:
1463:
1458:
1445:
1440:
1427:
1422:
1409:
1404:
1385:
1370:
1366:
1357:
1353:
1335:
1331:
1322:
1318:
1312:
1309:
1308:
1283:
1279:
1270:
1266:
1245:
1241:
1232:
1228:
1217:
1214:
1213:
1188:
1177:
1176:
1175:
1166:
1155:
1154:
1153:
1138:
1121:
1120:
1106:
1105:
1088:
1087:
1084:
1081:
1080:
1047:
1046:
1044:
1041:
1040:
1009:
1008:
994:
993:
973:
972:
958:
957:
952:
949:
948:
922:
921:
907:
906:
904:
901:
900:
871:
870:
856:
855:
854:
851:
837:
836:
828:
825:
824:
802:
799:
798:
776:
775:
761:
760:
758:
755:
754:
751:
727:
724:
723:
701:
698:
697:
681:
678:
677:
661:
658:
657:
641:
638:
637:
618:
615:
614:
592:
589:
588:
572:
569:
568:
566:
550:
539:
527:
519:
516:
515:
496:
493:
492:
470:
437:
432:
431:
420:
411:
406:
405:
394:
385:
380:
379:
368:
359:
354:
353:
342:
333:
328:
327:
316:
307:
302:
301:
290:
288:
285:
284:
265:
262:
261:
242:
231:
223:
212:
209:
206:
205:
179:
176:
175:
156:
153:
152:
129:
122:
50:
35:
28:
23:
22:
18:Angle bisectors
15:
12:
11:
5:
4752:
4742:
4741:
4714:
4713:
4694:
4684:
4678:
4672:
4666:
4655:
4654:External links
4652:
4650:
4649:
4636:
4624:
4609:
4567:
4554:
4535:
4523:
4508:
4493:
4480:
4464:
4442:
4436:Spain, Barry.
4429:
4417:
4415:
4412:
4406:
4403:
4391:
4388:
4379:
4376:
4374:
4371:
4354:
4351:
4342:
4339:
4281:
4275:
4272:
4266:
4263:
4260:
4257:
4254:
4249:
4245:
4238:
4235:
4190:
4187:
4184:
4181:
4176:
4142:
4139:
4137:
4134:
4118:(that is, has
4082:
4079:
4067:
4064:
4044:
4041:
4038:
4035:
4032:
4012:
4003:
3999:
3995:
3990:
3986:
3982:
3977:
3973:
3967:
3964:
3961:
3954:
3949:
3945:
3924:
3915:
3911:
3907:
3902:
3898:
3894:
3889:
3885:
3879:
3876:
3873:
3866:
3861:
3857:
3837:
3828:
3824:
3820:
3815:
3811:
3807:
3802:
3798:
3792:
3789:
3786:
3779:
3774:
3770:
3751:acute triangle
3731:Main article:
3728:
3725:
3720:center of mass
3703:
3700:
3698:
3695:
3693:
3690:
3670:Main article:
3667:
3664:
3650:
3647:
3638:
3635:
3617:either form a
3607:
3604:
3595:
3592:
3588:
3587:
3576:
3571:
3567:
3563:
3560:
3557:
3554:
3551:
3548:
3545:
3540:
3535:
3531:
3524:
3521:
3514:
3510:
3506:
3503:
3500:
3497:
3491:
3486:
3481:
3477:
3470:
3467:
3460:
3456:
3452:
3449:
3446:
3443:
3437:
3432:
3427:
3423:
3416:
3413:
3406:
3402:
3398:
3395:
3392:
3389:
3361:
3357:
3336:
3331:
3327:
3323:
3318:
3314:
3283:
3282:
3271:
3268:
3265:
3262:
3259:
3256:
3251:
3246:
3242:
3208:
3204:
3192:
3191:
3180:
3175:
3172:
3167:
3164:
3158:
3155:
3152:
3147:
3144:
3141:
3124:
3123:
3112:
3106:
3103:
3100:
3093:
3090:
3087:
3084:
3081:
3078:
3075:
3072:
3067:
3041:
3021:
3018:
3014:
3010:
3007:
3004:
3001:
2998:
2995:
2992:
2989:
2986:
2966:
2963:
2960:
2957:
2954:
2942:
2939:
2915:Main article:
2912:
2909:
2891:interior angle
2866:
2863:
2861:
2858:
2857:
2856:
2845:
2837:
2832:
2828:
2824:
2819:
2814:
2810:
2802:
2798:
2794:
2791:
2786:
2782:
2778:
2775:
2770:
2766:
2759:
2756:
2748:
2743:
2739:
2735:
2730:
2725:
2721:
2713:
2709:
2705:
2702:
2697:
2693:
2689:
2686:
2681:
2677:
2650:
2647:
2644:
2639:
2635:
2631:
2628:
2623:
2619:
2615:
2612:
2607:
2603:
2582:
2579:
2574:
2570:
2566:
2563:
2558:
2554:
2550:
2547:
2542:
2538:
2519:Pierre Wantzel
2482:angle bisector
2469:
2468:Angle bisector
2466:
2452:
2448:
2445:
2441:
2437:
2433:
2429:
2426:
2422:
2411:the property:
2400:
2380:
2377:
2346:
2342:
2337:
2332:
2328:
2324:
2319:
2314:
2310:
2306:
2301:
2296:
2292:
2288:
2283:
2278:
2274:
2270:
2265:
2260:
2256:
2252:
2247:
2242:
2238:
2234:
2228:
2225:
2219:
2216:
2213:
2208:
2204:
2200:
2195:
2191:
2187:
2184:
2181:
2178:
2173:
2169:
2165:
2160:
2156:
2152:
2149:
2146:
2143:
2138:
2134:
2130:
2125:
2121:
2117:
2091:
2086:
2082:
2078:
2073:
2069:
2065:
2060:
2056:
2052:
2049:
2046:
2043:
2040:
2035:
2031:
2027:
2022:
2018:
2014:
2009:
2005:
2001:
1998:
1995:
1973:
1970:
1965:
1958:
1955:
1948:
1943:
1936:
1933:
1926:
1920:
1917:
1911:
1908:
1902:
1899:
1893:
1887:
1884:
1878:
1875:
1869:
1866:
1846:
1845:
1828:
1825:
1822:Bisector plane
1816:
1815:
1809:
1808:of a triangle,
1802:
1799:Thales' circle
1790:
1787:
1773:
1768:
1764:
1760:
1755:
1751:
1747:
1741:
1738:
1732:
1727:
1723:
1700:
1695:
1691:
1687:
1682:
1678:
1674:
1668:
1665:
1659:
1654:
1650:
1622:
1618:
1614:
1609:
1605:
1597:
1593:
1589:
1584:
1580:
1572:
1569:
1566:
1541:
1537:
1533:
1530:
1525:
1521:
1517:
1514:
1511:
1508:
1505:
1502:
1488:Or explicitly:
1475:
1471:
1466:
1461:
1457:
1453:
1448:
1443:
1439:
1435:
1430:
1425:
1421:
1417:
1412:
1407:
1403:
1399:
1393:
1390:
1384:
1381:
1378:
1373:
1369:
1365:
1360:
1356:
1352:
1349:
1346:
1343:
1338:
1334:
1330:
1325:
1321:
1317:
1291:
1286:
1282:
1278:
1273:
1269:
1265:
1262:
1259:
1256:
1253:
1248:
1244:
1240:
1235:
1231:
1227:
1224:
1221:
1199:
1196:
1191:
1184:
1181:
1174:
1169:
1162:
1159:
1152:
1146:
1143:
1137:
1134:
1128:
1125:
1119:
1113:
1110:
1104:
1101:
1095:
1092:
1063:
1060:
1054:
1051:
1028:
1025:
1022:
1016:
1013:
1007:
1001:
998:
992:
989:
986:
980:
977:
971:
965:
962:
956:
929:
926:
920:
914:
911:
885:
878:
875:
869:
863:
860:
850:
844:
841:
835:
832:
812:
809:
806:
783:
780:
774:
768:
765:
750:
747:
734:
731:
711:
708:
705:
696:in two points
685:
665:
645:
622:
596:
576:
553:
549:
546:
542:
535:
532:
526:
523:
503:
500:
469:
466:
458:
457:
446:
440:
435:
430:
427:
423:
419:
414:
409:
404:
401:
397:
393:
388:
383:
378:
375:
371:
367:
362:
357:
352:
349:
345:
341:
336:
331:
326:
323:
319:
315:
310:
305:
300:
297:
293:
245:
241:
238:
234:
230:
226:
222:
219:
215:
200:
199:
183:
163:
160:
149:
128:
125:
121:
118:
94:angle bisector
26:
9:
6:
4:
3:
2:
4751:
4740:
4737:
4736:
4734:
4727:
4726:
4724:
4720:
4709:
4708:
4703:
4700:
4695:
4692:
4688:
4685:
4682:
4679:
4676:
4673:
4670:
4667:
4665:
4661:
4658:
4657:
4646:
4640:
4634:
4628:
4621:
4620:
4613:
4605:
4601:
4597:
4593:
4589:
4585:
4578:
4576:
4574:
4572:
4564:
4558:
4552:
4548:
4542:
4540:
4533:
4527:
4521:
4519:
4512:
4505:
4504:
4497:
4490:
4484:
4478:
4474:
4468:
4461:
4455:
4453:
4451:
4449:
4447:
4439:
4433:
4427:
4422:
4418:
4411:
4402:
4401:
4397:
4390:Quadrilateral
4387:
4385:
4378:Parallelogram
4370:
4368:
4364:
4360:
4350:
4348:
4347:parallelogram
4341:Parallelogram
4338:
4336:
4331:
4329:
4325:
4320:
4318:
4314:
4310:
4306:
4302:
4298:
4293:
4279:
4273:
4270:
4264:
4258:
4252:
4247:
4243:
4236:
4233:
4222:
4218:
4214:
4211:
4207:
4202:
4188:
4185:
4182:
4179:
4174:
4164:
4160:
4156:
4152:
4148:
4133:
4131:
4126:
4124:
4121:
4120:perpendicular
4117:
4116:orthodiagonal
4113:
4109:
4107:
4103:
4098:
4095:
4094:quadrilateral
4092:
4088:
4081:Quadrilateral
4078:
4065:
4062:
4042:
4039:
4036:
4033:
4030:
4010:
4001:
3997:
3993:
3988:
3984:
3980:
3975:
3971:
3965:
3962:
3959:
3952:
3947:
3943:
3922:
3913:
3909:
3905:
3900:
3896:
3892:
3887:
3883:
3877:
3874:
3871:
3864:
3859:
3855:
3835:
3826:
3822:
3818:
3813:
3809:
3805:
3800:
3796:
3790:
3787:
3784:
3777:
3772:
3768:
3758:
3756:
3752:
3747:
3745:
3740:
3739:perpendicular
3737:The interior
3734:
3724:
3721:
3717:
3713:
3709:
3689:
3687:
3686:perpendicular
3683:
3679:
3673:
3663:
3661:
3656:
3646:
3644:
3634:
3632:
3628:
3624:
3620:
3616:
3615:quadrilateral
3613:
3606:Quadrilateral
3603:
3601:
3591:
3574:
3569:
3561:
3558:
3555:
3552:
3549:
3543:
3538:
3533:
3529:
3522:
3519:
3512:
3504:
3501:
3498:
3489:
3484:
3479:
3475:
3468:
3465:
3458:
3450:
3447:
3444:
3435:
3430:
3425:
3421:
3414:
3411:
3404:
3396:
3393:
3390:
3377:
3376:
3375:
3359:
3355:
3334:
3329:
3325:
3321:
3316:
3312:
3302:
3300:
3296:
3292:
3288:
3269:
3266:
3263:
3260:
3257:
3254:
3249:
3244:
3240:
3232:
3231:
3230:
3228:
3224:
3206:
3202:
3178:
3173:
3170:
3165:
3162:
3156:
3153:
3150:
3145:
3142:
3139:
3129:
3128:
3127:
3110:
3104:
3101:
3098:
3088:
3085:
3082:
3076:
3073:
3070:
3065:
3055:
3054:
3053:
3039:
3019:
3016:
3012:
3005:
3002:
2999:
2996:
2993:
2987:
2984:
2964:
2961:
2958:
2955:
2952:
2938:
2936:
2932:
2923:
2918:
2908:
2905:
2903:
2899:
2898:extended side
2894:
2892:
2888:
2880:
2876:
2871:
2843:
2835:
2830:
2826:
2822:
2817:
2812:
2808:
2800:
2796:
2792:
2789:
2784:
2780:
2776:
2773:
2768:
2764:
2757:
2754:
2746:
2741:
2737:
2733:
2728:
2723:
2719:
2711:
2707:
2703:
2700:
2695:
2691:
2687:
2684:
2679:
2675:
2664:
2663:
2662:
2648:
2645:
2642:
2637:
2633:
2629:
2626:
2621:
2617:
2613:
2610:
2605:
2601:
2580:
2577:
2572:
2568:
2564:
2561:
2556:
2552:
2548:
2545:
2540:
2536:
2527:
2526:perpendicular
2522:
2520:
2516:
2511:
2509:
2504:
2502:
2498:
2493:
2491:
2487:
2483:
2474:
2465:
2446:
2443:
2435:
2427:
2424:
2398:
2378:
2375:
2367:
2362:
2357:
2344:
2335:
2330:
2326:
2322:
2317:
2312:
2308:
2304:
2299:
2294:
2290:
2286:
2281:
2276:
2272:
2268:
2263:
2258:
2254:
2250:
2245:
2240:
2236:
2226:
2223:
2217:
2214:
2206:
2202:
2198:
2193:
2189:
2182:
2179:
2171:
2167:
2163:
2158:
2154:
2147:
2144:
2136:
2132:
2128:
2123:
2119:
2107:
2103:
2084:
2080:
2076:
2071:
2067:
2063:
2058:
2054:
2047:
2044:
2041:
2033:
2029:
2025:
2020:
2016:
2012:
2007:
2003:
1996:
1993:
1984:
1971:
1963:
1953:
1946:
1941:
1931:
1918:
1915:
1909:
1897:
1891:
1882:
1873:
1864:
1853:
1849:
1843:
1839:
1835:
1834:perpendicular
1831:
1830:
1820:
1813:
1810:
1807:
1803:
1800:
1796:
1795:
1794:
1786:
1766:
1762:
1758:
1753:
1749:
1739:
1736:
1730:
1725:
1721:
1693:
1689:
1685:
1680:
1676:
1666:
1663:
1657:
1652:
1648:
1620:
1616:
1612:
1607:
1603:
1595:
1591:
1587:
1582:
1578:
1570:
1567:
1564:
1539:
1535:
1531:
1523:
1519:
1515:
1512:
1506:
1503:
1500:
1492:
1486:
1473:
1464:
1459:
1455:
1451:
1446:
1441:
1437:
1433:
1428:
1423:
1419:
1415:
1410:
1405:
1401:
1391:
1388:
1382:
1379:
1371:
1367:
1363:
1358:
1354:
1347:
1344:
1336:
1332:
1328:
1323:
1319:
1307:
1303:
1284:
1280:
1276:
1271:
1267:
1260:
1257:
1254:
1246:
1242:
1238:
1233:
1229:
1222:
1219:
1210:
1197:
1189:
1179:
1172:
1167:
1157:
1144:
1141:
1135:
1123:
1117:
1108:
1099:
1090:
1079:
1075:
1061:
1058:
1049:
1039:. Inserting
1026:
1023:
1011:
1005:
996:
987:
975:
969:
960:
946:
945:normal vector
924:
918:
909:
883:
873:
867:
858:
848:
839:
833:
830:
810:
807:
804:
778:
772:
763:
746:
732:
729:
709:
706:
703:
683:
663:
643:
636:
620:
613:
608:
594:
574:
547:
544:
533:
530:
524:
521:
501:
498:
489:
487:
483:
474:
465:
463:
444:
438:
428:
425:
417:
412:
402:
399:
391:
386:
376:
373:
365:
360:
350:
347:
339:
334:
324:
321:
313:
308:
298:
295:
283:
282:
281:
279:
258:
239:
236:
228:
220:
217:
204:
197:
181:
161:
158:
150:
147:
143:
142:perpendicular
139:
138:
133:
124:
117:
115:
111:
107:
103:
99:
95:
91:
87:
83:
79:
75:
71:
67:
63:
54:
48:
44:
40:
33:
19:
4716:
4715:
4705:
4664:cut-the-knot
4644:
4639:
4627:
4617:
4612:
4587:
4583:
4562:
4557:
4546:
4526:
4517:
4511:
4501:
4496:
4488:
4483:
4472:
4467:
4459:
4437:
4432:
4421:
4408:
4393:
4381:
4356:
4344:
4332:
4321:
4294:
4203:
4144:
4127:
4110:
4099:
4084:
3759:
3748:
3744:circumcenter
3736:
3733:Circumcircle
3705:
3675:
3652:
3640:
3609:
3598:There exist
3597:
3589:
3303:
3298:
3294:
3290:
3286:
3284:
3226:
3222:
3193:
3125:
2944:
2928:
2906:
2895:
2884:
2523:
2512:
2505:
2494:
2484:divides the
2481:
2479:
2365:
2360:
2358:
2105:
2104:
1985:
1851:
1850:
1847:
1837:
1792:
1789:Applications
1490:
1487:
1305:
1304:
1211:
1077:
1076:
752:
634:
611:
609:
491:The segment
490:
479:
461:
459:
259:
202:
201:
123:
113:
93:
81:
77:
65:
59:
4549:13, 53-59.
4396:Newton Line
4328:Nagel point
3662:intersect.
899:and vector
635:given point
196:equidistant
88:of a given
4719:PlanetMath
4414:References
4361:, and any
4221:asymptotic
4217:hyperbolas
4159:concurrent
4106:concurrent
3627:concurrent
2875:concurrent
127:Definition
92:, and the
32:Dissection
4707:MathWorld
4384:diagonals
4367:diameters
4301:perimeter
4265:−
4253:
4219:that are
4123:diagonals
4087:bimedians
4040:≥
4034:≥
3981:−
3906:−
3819:−
3623:concyclic
3166:
3086:−
2902:collinear
2758:±
2497:half-line
2359:Property
2323:−
2287:−
2251:−
2199:−
2164:−
2129:−
1957:→
1947:−
1935:→
1901:→
1892:−
1886:→
1874:⋅
1868:→
1613:−
1588:−
1571:−
1516:−
1452:−
1416:−
1364:−
1329:−
1183:→
1173:−
1161:→
1127:→
1118:−
1112:→
1100:⋅
1094:→
1062:⋯
1053:→
1015:→
1006:−
1000:→
988:⋅
979:→
970:−
964:→
928:→
919:−
913:→
877:→
862:→
843:→
782:→
767:→
749:Equations
460:Property
70:congruent
66:bisection
4733:Category
4335:incircle
4324:splitter
4313:incircle
4206:envelope
4163:centroid
4151:triangle
4141:Triangle
4085:The two
3716:centroid
3697:Triangle
3682:parabola
3666:Parabola
2935:triangle
2879:incenter
2860:Triangle
1842:midpoint
1806:Excircle
146:midpoint
114:bisector
86:midpoint
78:bisector
62:geometry
4604:3615256
4315:of the
4297:cleaver
4210:deltoid
4155:medians
3708:medians
3702:Medians
3678:tangent
3643:rhombus
3637:Rhombus
3374:, then
3229:, then
2941:Lengths
2931:lengths
90:segment
4602:
4363:chords
4359:center
4305:concur
4102:cyclic
4091:convex
3749:In an
3712:vertex
3612:convex
3285:where
2900:, are
1712:, and
1556:where
100:of an
4600:JSTOR
4149:of a
4089:of a
3680:to a
2490:equal
2486:angle
1986:With
1838:plane
1212:With
943:is a
633:at a
110:plane
102:angle
4689:and
4382:The
4204:The
4147:area
4128:The
3935:and
3676:The
3347:and
3289:and
3225:and
2593:and
2106:(C3)
1832:The
525:>
486:arcs
276:and
140:The
98:apex
74:line
4662:at
4592:doi
4244:log
3163:cos
2521:).
2480:An
2366:(D)
2361:(D)
1852:(V)
1491:(E)
1306:(C)
1078:(V)
753:If
462:(D)
203:(D)
194:is
60:In
4735::
4704:.
4598:.
4588:56
4586:.
4570:^
4538:^
4445:^
4322:A
4295:A
3633:.
3602:.
3301:.
2464:.
1785:.
1639:,
1554:,
745:.
280::
257:.
116:.
64:,
4725:.
4710:.
4606:.
4594::
4280:,
4274:2
4271:1
4262:)
4259:2
4256:(
4248:e
4237:4
4234:3
4189:1
4186::
4183:1
4180:+
4175:2
4066:.
4063:T
4043:c
4037:b
4031:a
4011:,
4002:2
3998:c
3994:+
3989:2
3985:b
3976:2
3972:a
3966:T
3963:c
3960:2
3953:=
3948:c
3944:p
3923:,
3914:2
3910:c
3901:2
3897:b
3893:+
3888:2
3884:a
3878:T
3875:b
3872:2
3865:=
3860:b
3856:p
3836:,
3827:2
3823:c
3814:2
3810:b
3806:+
3801:2
3797:a
3791:T
3788:a
3785:2
3778:=
3773:a
3769:p
3575:.
3570:2
3566:)
3562:c
3559:+
3556:b
3553:+
3550:a
3547:(
3544:=
3539:2
3534:c
3530:t
3523:b
3520:a
3513:2
3509:)
3505:b
3502:+
3499:a
3496:(
3490:+
3485:2
3480:b
3476:t
3469:a
3466:c
3459:2
3455:)
3451:a
3448:+
3445:c
3442:(
3436:+
3431:2
3426:a
3422:t
3415:c
3412:b
3405:2
3401:)
3397:c
3394:+
3391:b
3388:(
3360:c
3356:t
3335:,
3330:b
3326:t
3322:,
3317:a
3313:t
3299:c
3297::
3295:b
3291:c
3287:b
3270:c
3267:b
3264:=
3261:n
3258:m
3255:+
3250:2
3245:a
3241:t
3227:n
3223:m
3207:a
3203:t
3179:.
3174:2
3171:A
3157:c
3154:+
3151:b
3146:c
3143:b
3140:2
3111:,
3105:c
3102:+
3099:b
3092:)
3089:a
3083:s
3080:(
3077:s
3074:c
3071:b
3066:2
3040:a
3020:,
3017:2
3013:/
3009:)
3006:c
3003:+
3000:b
2997:+
2994:a
2991:(
2988:=
2985:s
2965:c
2962:,
2959:b
2956:,
2953:a
2844:.
2836:2
2831:2
2827:m
2823:+
2818:2
2813:2
2809:l
2801:2
2797:n
2793:+
2790:y
2785:2
2781:m
2777:+
2774:x
2769:2
2765:l
2755:=
2747:2
2742:1
2738:m
2734:+
2729:2
2724:1
2720:l
2712:1
2708:n
2704:+
2701:y
2696:1
2692:m
2688:+
2685:x
2680:1
2676:l
2649:,
2646:0
2643:=
2638:2
2634:n
2630:+
2627:y
2622:2
2618:m
2614:+
2611:x
2606:2
2602:l
2581:0
2578:=
2573:1
2569:n
2565:+
2562:y
2557:1
2553:m
2549:+
2546:x
2541:1
2537:l
2451:|
2447:B
2444:X
2440:|
2436:=
2432:|
2428:A
2425:X
2421:|
2399:X
2379:B
2376:A
2345:.
2341:)
2336:2
2331:3
2327:b
2318:2
2313:3
2309:a
2305:+
2300:2
2295:2
2291:b
2282:2
2277:2
2273:a
2269:+
2264:2
2259:1
2255:b
2246:2
2241:1
2237:a
2233:(
2227:2
2224:1
2218:=
2215:z
2212:)
2207:3
2203:b
2194:3
2190:a
2186:(
2183:+
2180:y
2177:)
2172:2
2168:b
2159:2
2155:a
2151:(
2148:+
2145:x
2142:)
2137:1
2133:b
2124:1
2120:a
2116:(
2090:)
2085:3
2081:b
2077:,
2072:2
2068:b
2064:,
2059:1
2055:b
2051:(
2048:=
2045:B
2042:,
2039:)
2034:3
2030:a
2026:,
2021:2
2017:a
2013:,
2008:1
2004:a
2000:(
1997:=
1994:A
1972:.
1969:)
1964:2
1954:b
1942:2
1932:a
1925:(
1919:2
1916:1
1910:=
1907:)
1898:b
1883:a
1877:(
1865:x
1801:,
1772:)
1767:2
1763:b
1759:+
1754:2
1750:a
1746:(
1740:2
1737:1
1731:=
1726:0
1722:y
1699:)
1694:1
1690:b
1686:+
1681:1
1677:a
1673:(
1667:2
1664:1
1658:=
1653:0
1649:x
1621:2
1617:a
1608:2
1604:b
1596:1
1592:a
1583:1
1579:b
1568:=
1565:m
1540:0
1536:y
1532:+
1529:)
1524:0
1520:x
1513:x
1510:(
1507:m
1504:=
1501:y
1474:.
1470:)
1465:2
1460:2
1456:b
1447:2
1442:2
1438:a
1434:+
1429:2
1424:1
1420:b
1411:2
1406:1
1402:a
1398:(
1392:2
1389:1
1383:=
1380:y
1377:)
1372:2
1368:b
1359:2
1355:a
1351:(
1348:+
1345:x
1342:)
1337:1
1333:b
1324:1
1320:a
1316:(
1290:)
1285:2
1281:b
1277:,
1272:1
1268:b
1264:(
1261:=
1258:B
1255:,
1252:)
1247:2
1243:a
1239:,
1234:1
1230:a
1226:(
1223:=
1220:A
1198:.
1195:)
1190:2
1180:b
1168:2
1158:a
1151:(
1145:2
1142:1
1136:=
1133:)
1124:b
1109:a
1103:(
1091:x
1059:=
1050:m
1027:0
1024:=
1021:)
1012:b
997:a
991:(
985:)
976:m
961:x
955:(
925:b
910:a
884:2
874:b
868:+
859:a
849:=
840:m
834::
831:M
811:B
808:,
805:A
779:b
773:,
764:a
733:B
730:A
710:B
707:,
704:A
684:g
664:P
644:P
621:g
595:M
575:M
552:|
548:B
545:A
541:|
534:2
531:1
522:r
502:B
499:A
445:.
439:2
434:|
429:B
426:X
422:|
418:=
413:2
408:|
403:B
400:M
396:|
392:+
387:2
382:|
377:M
374:X
370:|
366:=
361:2
356:|
351:A
348:M
344:|
340:+
335:2
330:|
325:M
322:X
318:|
314:=
309:2
304:|
299:A
296:X
292:|
244:|
240:B
237:X
233:|
229:=
225:|
221:A
218:X
214:|
182:X
162:B
159:A
49:.
34:.
20:)
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