1772:
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437:"If a visitor from Mars desired to learn the geometry of the triangle but could stay in the earth's relatively dense atmosphere only long enough for a single lesson, earthling mathematicians would, no doubt, be hard-pressed to meet this request. In this paper, we believe that we have an optimum solution to the problem. The Kiepert conics ...."
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433:. The following quote from a paper by R. H. Eddy and R. Fritsch is enough testimony to establish the importance of the Kiepert conics in the study of triangle geometry:
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in 1868: "Construct a triangle, given the peaks of the equilateral triangles constructed on the sides." A solution to the problem was published by
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1386:
740:
1777:
Kiepert parabola of triangle ABC. The figure also shows a member (line LMN) of the family of lines whose envelope is the
Kiepert parabola.
1241:
458:
in 1869 and the solution contained a remark which effectively stated the locus definition of the
Kiepert hyperbola alluded to earlier.
1783:
455:
447:
1903:
Eddy, R. H.; Fritsch, R. (1994). "The Conics of Ludwig
Kiepert: A Comprehensive Lesson in the Geometry of the Triangle".
349:
205:
1191:
The
Kiepert parabola was first studied in 1888 by a German mathematics teacher Augustus Artzt in a "school program".
733:
The centre of the
Kiepert hyperbola is the triangle center X(115). The barycentric coordinates of the center are
1542:
The focus of the
Kiepert parabola is the triangle center X(110). The barycentric coordinates of the focus are
1971:
422:
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120:
87:
54:
20:
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405:
It has been proved that the
Kiepert hyperbola is the hyperbola passing through the vertices, the
317:
1873:
394:
250:
889:
717:{\displaystyle {\frac {b^{2}-c^{2}}{x}}+{\frac {c^{2}-a^{2}}{y}}+{\frac {a^{2}-b^{2}}{z}}=0.}
398:
256:
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930:. The center is the midpoint of the line segment joining the isogonic centers of triangle
8:
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Kiepert hyperbola showing the orthocenter, the incenter and the perpendicular asymptotes
1947:
1139:
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which are the triangle centers X(13) and X(14) in the
Encyclopedia of Triangle Centers.
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1939:
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430:
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Kiepert hyperbola showing the center of perspectivity of triangles ABC and A'B'C'
994:
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1965:
451:
176:
as bases, are similar, isosceles and similarly situated, then the triangles
1696:{\displaystyle a^{2}/(b^{2}-c^{2}):b^{2}/(c^{2}-a^{2}):c^{2}/(a^{2}-b^{2})}
998:
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while investigating the solution of the following problem proposed by
1528:{\displaystyle f=(b^{2}-c^{2})/a,g=(c^{2}-a^{2})/b,h=(a^{2}-b^{2})/c}
861:{\displaystyle (b^{2}-c^{2})^{2}:(c^{2}-a^{2})^{2}:(a^{2}-b^{2})^{2}}
32:
1709:
The directrix of the
Kiepert parabola is the Euler line of triangle
1374:{\displaystyle f^{2}x^{2}+g^{2}y^{2}+h^{2}z^{2}-2fgxy-2ghyz-2hfzx=0}
413:
of the reference triangle and the
Kiepert parabola is the parabola
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40:
564:
The equation of the Kiepert hyperbola in barycentric coordinates
414:
1200:
The equation of the Kiepert parabola in barycentric coordinates
253:. As the base angle of the isosceles triangles varies between
28:
31:
associated with the reference triangle. One of them is a
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be a point in the plane of a nonequilateral triangle
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Kiepert parabola showing the focus and the directrix
393:
is a hyperbola called the Kiepert hyperbola and the
1930:Sharp, J. (2015). "Artzt parabolas of a triangle".
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386:{\displaystyle A^{\prime }B^{\prime }C^{\prime }}
242:{\displaystyle A^{\prime }B^{\prime }C^{\prime }}
1963:
926:The center of the Kiepert hyperbola lies on the
873:The asymptotes of the Kiepert hyperbola are the
1837:
1817:
959:The image of the Kiepert hyperbola under the
47:. The Kiepert conics are defined as follows:
530:the vertex angles of the reference triangle
1902:
1871:
401:is a parabola called the Kiepert parabola.
1929:
150:, constructed on the sides of a triangle
446:The Kiepert hyperbola was discovered by
16:Conic curves associated with a triangle
1964:
1898:
1896:
1894:
1156:is perpendicular to the Euler line of
727:
417:in the reference triangle having the
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13:
1874:"X(110)=Focus of Kiepert Parabola"
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135:
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63:
14:
1983:
1844:MathWorld--A Wolfram Web Resource
1824:MathWorld--A Wolfram Web Resource
1811:
1878:Encyclopedia of Triangle Centers
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1770:
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1194:
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1917:10.1080/0025570X.1994.11996212
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993:which is the line joining the
892:and hence its eccentricity is
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1:
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143:{\displaystyle ABC^{\prime }}
110:{\displaystyle AB^{\prime }C}
77:{\displaystyle A^{\prime }BC}
877:of the intersections of the
7:
1794:
909:{\displaystyle {\sqrt {2}}}
888:The Kiepert hyperbola is a
559:
10:
1988:
1116:. The locus of the points
1070:be the trilinear polar of
1739:
1182:is the Kiepert hyperbola.
425:and the triangle center X
1932:The Mathematical Gazette
1806:Modern triangle geometry
498:be the side lengths and
961:isogonal transformation
318:center of perspectivity
277:{\displaystyle -\pi /2}
51:If the three triangles
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305:{\displaystyle \pi /2}
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1225:{\displaystyle x:y:z}
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890:rectangular hyperbola
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589:{\displaystyle x:y:z}
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523:{\displaystyle A,B,C}
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491:{\displaystyle a,b,c}
399:axis of perspectivity
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1944:10.1017/mag.2015.81
1838:Weisstein, Eric W.
1820:"Kiepert Hyperbola"
1818:Weisstein, Eric W.
1728:{\displaystyle ABC}
1175:{\displaystyle ABC}
1109:{\displaystyle ABC}
1043:{\displaystyle ABC}
986:{\displaystyle ABC}
949:{\displaystyle ABC}
549:{\displaystyle ABC}
339:{\displaystyle ABC}
195:{\displaystyle ABC}
169:{\displaystyle ABC}
39:and the other is a
1840:"Kiepert Parabola"
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728:Center, asymptotes
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1972:Triangle geometry
1149:{\displaystyle p}
1129:{\displaystyle P}
1083:{\displaystyle P}
1063:{\displaystyle p}
1017:{\displaystyle P}
928:nine-point circle
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442:Kiepert hyperbola
320:of the triangles
37:Kiepert hyperbola
21:triangle geometry
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1938:(546): 444–463.
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1187:Kiepert parabola
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45:Kiepert parabola
27:are two special
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995:symmedian point
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1911:(3): 188–205.
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1812:External links
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1801:Triangle conic
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456:Ludvig Kiepert
448:Ludvig Kiepert
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25:Kiepert conics
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