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Geometric transformation applied to points with respect to a given triangle
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Interactive Java Applet illustrating isogonal conjugate and its properties
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in 0, 1, or 2 points. The isogonal conjugate of the circumcircle is the
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Isogonal conjugate transformation over the points inside the triangle.
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326:{\displaystyle {\tfrac {1}{x}}:{\tfrac {1}{y}}:{\tfrac {1}{z}}.}
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given a generalization of isogonal conjugate as follows: Let
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of triangle centers under the trilinear product, defined by
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Another construction for the isogonal conjugate of a point
887:, Dao's generalization become the isogonal conjugate of
146:.) This is a direct result of the trigonometric form of
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61:reflected about the angle bisectors (concur at
1008:César Eliud Lozada, Preamble before X(44687)
333:For this reason, the isogonal conjugate of
127:respectively. These three reflected lines
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531:A second definition of isogonal conjugate
438:{\displaystyle (p:q:r)*(u:v:w)=pu:qv:rw,}
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850:{\displaystyle D=D(X,P)=x*(x-y-z)*q*r::}
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219:are isogonal conjugates of each other.
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1036:Pedal Triangle and Isogonal Conjugacy
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489:according as the line intersects the
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980:"Constructing Isogonal Conjugates"
906:, Dao's generalization become the
153:The isogonal conjugate of a point
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271:, then its isogonal conjugate is
207:. The isogonal conjugates of the
1011:Encyclopedia of Triangle Centers
925:Dao's generalization become the
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452:, and the inverse of each
666:respectively. Then lines
619:a point on its plane and
546:, let the reflections of
539:in the plane of triangle
718:{\displaystyle X=x:y:z:}
337:is sometimes denoted by
157:is sometimes denoted by
959:Central line (geometry)
921:is the circumconic of
756:{\displaystyle P=p:q:r}
257:{\displaystyle X=x:y:z}
200:is (by definition) the
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215:and vice versa. The
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668:AA", BB", CC"
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623:an arbitrary
610:
609:Dao Thanh Oai
607:In may 2021,
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507:Neuberg cubic
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988:. Retrieved
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139:of triangle
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625:circumconic
475:circumconic
181:orthocentre
990:17 January
965:References
859:The point
664:A", B", C"
656:BC, CA, AB
641:A', B', C'
635:cut again
633:AP, BP, CP
552:BC, CA, AB
119:about the
117:PA, PB, PC
115:the lines
113:reflecting
1031:MathWorld
839:∗
833:∗
824:−
818:−
809:∗
671:concurent
662:again at
487:hyperbola
382:∗
1045:Category
984:GeoGebra
938:See also
645:parallel
631:. Lines
483:parabola
471:function
211:are the
195:centroid
174:incentre
137:sideline
103:triangle
88:geometry
37:incenter
898:is the
879:is the
763:, then
501:(e.g.,
479:ellipse
343:. The
186:is the
125:A, B, C
652:points
499:cubics
129:concur
90:, the
65:, the
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33:concur
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917:When
894:When
875:When
648:lines
629:△ ABC
613:△ ABC
485:, or
448:is a
226:, if
96:point
94:of a
992:2022
771:is:
725:and
684:are
658:cut
929:of
923:ABC
910:of
904:ABC
902:of
885:ABC
883:of
680:of
654:to
639:at
627:of
564:, P
560:, P
554:be
543:ABC
460:is
456:in
345:set
268:ABC
222:In
165:is
143:ABC
123:of
108:ABC
86:In
69:of
35:at
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1000:^
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673:.
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169:.
163:P*
159:P*
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994:.
933:.
931:P
919:Ω
914:.
912:P
896:Ω
891:.
889:P
877:Ω
869:P
865:X
861:D
842:r
836:q
830:)
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812:(
806:x
803:=
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782:=
779:D
765:D
751:r
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736:=
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710:z
707::
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695:=
692:X
682:Ω
678:X
660:Ω
637:Ω
621:Ω
617:P
589:P
583:c
581:P
579:b
577:P
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573:P
571:〇
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541:△
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427:r
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40:I
31:(
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