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- Isogonal_conjugate abstract "In geometry, the isogonal conjugate of a point P with respect to a triangle ABC is constructed by reflecting the lines PA, PB, and PC about the angle bisectors of A, B, and C. These three reflected lines concur at the isogonal conjugate of P. (This definition applies only to points not on a sideline of triangle ABC.)The isogonal conjugate of a point P is sometimes denoted by P*. The isogonal conjugate of P* is P.The isogonal conjugate of the incentre I is itself. The isogonal conjugate of the orthocentre H is the circumcentre O. The isogonal conjugate of the centroid G is (by definition) the symmedian point K. The isogonal conjugates of the Fermat points are the isodynamic points and vice versa. The Brocard points are isogonal conjugates of each other.In trilinear coordinates, if X = x : y : z is a point not on a sideline of triangle ABC, then its isogonal conjugate is 1/x : 1/y : 1/z. For this reason, the isogonal conjugate of X is sometimes denoted by X −1. The set S of triangle centers under trilinear product, defined by (p : q : r) * (u : v : w) = pu : qv : rw,is a commutative group, and the inverse of each X in S is X −1.As isogonal conjugation is a function, it makes sense to speak of the isogonal conjugate of sets of points, such as lines and circles. For example, the isogonal conjugate of a line is a circumconic; specifically, an ellipse, parabola, or hyperbola according as the line intersects the circumcircle in 0, 1, or 2 points. The isogonal conjugate of the circumcircle is the line at infinity. Several well-known cubics (e.g., Thompson cubic, Darboux cubic, Neuberg cubic) are self-isogonal-conjugate, in the sense that if X is on the cubic, then X −1 is also on the cubic.".
- Isogonal_conjugate thumbnail Isogonal_Conjugate.svg?width=300.
- Isogonal_conjugate wikiPageExternalLink IsogonalConjugate.html.
- Isogonal_conjugate wikiPageExternalLink OrthologicPedal.shtml.
- Isogonal_conjugate wikiPageExternalLink isogonal-conjugate_en.html.
- Isogonal_conjugate wikiPageID "2083100".
- Isogonal_conjugate wikiPageRevisionID "540444816".
- Isogonal_conjugate hasPhotoCollection Isogonal_conjugate.
- Isogonal_conjugate subject Category:Triangle_geometry.
- Isogonal_conjugate comment "In geometry, the isogonal conjugate of a point P with respect to a triangle ABC is constructed by reflecting the lines PA, PB, and PC about the angle bisectors of A, B, and C. These three reflected lines concur at the isogonal conjugate of P. (This definition applies only to points not on a sideline of triangle ABC.)The isogonal conjugate of a point P is sometimes denoted by P*. The isogonal conjugate of P* is P.The isogonal conjugate of the incentre I is itself.".
- Isogonal_conjugate label "Coniugato isogonale".
- Isogonal_conjugate label "Conjugado isogonal".
- Isogonal_conjugate label "Conjugué isogonal".
- Isogonal_conjugate label "Isogonal conjugate".
- Isogonal_conjugate label "Isogonal konjugierte Punkte".
- Isogonal_conjugate label "Isogonale verwantschap".
- Isogonal_conjugate label "Sprzężenie izogonalne".
- Isogonal_conjugate label "Изогональное сопряжение".
- Isogonal_conjugate label "等角共轭".
- Isogonal_conjugate sameAs Isogonal_konjugierte_Punkte.
- Isogonal_conjugate sameAs Conjugado_isogonal.
- Isogonal_conjugate sameAs Conjugué_isogonal.
- Isogonal_conjugate sameAs Coniugato_isogonale.
- Isogonal_conjugate sameAs 등각켤레점.
- Isogonal_conjugate sameAs Isogonale_verwantschap.
- Isogonal_conjugate sameAs Sprzężenie_izogonalne.
- Isogonal_conjugate sameAs m.06l05n.
- Isogonal_conjugate sameAs Q1156867.
- Isogonal_conjugate sameAs Q1156867.
- Isogonal_conjugate wasDerivedFrom Isogonal_conjugate?oldid=540444816.
- Isogonal_conjugate depiction Isogonal_Conjugate.svg.
- Isogonal_conjugate isPrimaryTopicOf Isogonal_conjugate.
