The trilinear pole of the orthotransversal of a point P is called its orthocorrespondent.

The orthocorrespondent of a point P=p:q:r is given by


where S_A, S_B, and S_C is Conway triangle notation.

In general, there are two (not necessarily real) points sharing the same orthocorrespondent. These points are inverse in the polar circle. However, all points at infinity have the triangle centroid as their orthocorrespondent.

The following table gives the orthocorrespondents of finite Kimberling centers whose orthocorrespondents are also Kimberling centers.

X_1incenterX_(57)isogonal conjugate of X_9
X_2triangle centroidX_(1992)orthocorrespondent of X_2
X_3circumcenterX_(1993)orthocorrespondent of X_3
X_5nine-point centerX_(1994)orthocorrespondent of X_5
X_6symmedian pointX_(1995)orthocorrespondent of X_6
X_7Gergonne pointX_(1996)orthocorrespondent of X_7
X_8Nagel pointX_(1997)orthocorrespondent of X_8
X_9mittenpunktX_(1998)orthocorrespondent of X_9
X_(10)Spieker centerX_(1999)orthocorrespondent of X_(10)
X_(11)Feuerbach pointX_(651)trilinear pole of line X_1X_3
X_(13)first Fermat pointX_(13)first Fermat point
X_(14)second Fermat pointX_(14)second Fermat point
X_(15)first isodynamic pointX_(62)isogonal conjugate of X_(18)
X_(16)second isodynamic pointX_(61)isogonal conjugate of X_(17)
X_(19)Clawson pointX_(2000)orthocorrespondent of X_(19)
X_(32)third power pointX_(2001)orthocorrespondent of X_(32)
X_(33)perspector of the orthic and intangents trianglesX_(2002)orthocorrespondent of X_(33)
X_(36)inverse-in-circumcircle of incenterX_(2003)orthocorrespondent of X_(36)
X_(61)isogonal conjugate of X_(17)X_(2004)orthocorrespondent of X_(61)
X_(62)isogonal conjugate of X_(18)X_(2005)orthocorrespondent of X_(62)
X_(80)reflection of incenter in Feuerbach pointX_(2006)orthocorrespondent of X_(80)
X_(98)Tarry pointX_(287)X_2-Hirst inverse of X_(98)
X_(100)anticomplement of Feuerbach pointX_(1332)orthocorrespondent of X_(100)
X_(101)psi(incenter, symmedian point)X_(1331)orthocorrespondent of X_(101)
X_(103)antipode of X_(101)X_(1815)inverse Mimosa transform of X_(910)
X_(105)lambda(incenter, symmedian point)X_(1814)inverse Mimosa transform of X_(672)
X_(106)lambda(incenter, triangle centroid)X_(1797)inverse Mimosa transform of X_(44)
X_(107)psi(symmedian point, orthocenter)X_(648)trilinear pole of Euler line
X_(108)psi(circumcenter, orthocenter)X_(651)trilinear pole of line X_1X_3
X_(109)psi(incenter, circumcenter)X_(1813)inverse Mimosa transform of X_(650)
X_(111)Parry pointX_(895)isogonal conjugate of X_(468)
X_(112)psi(orthocenter, symmedian point)X_(110)focus of Kiepert parabola
X_(113)Jerabek antipodeX_(2986)Dc(X_(74))
X_(114)Kiepert antipodeX_(2987)Dc(X_(98))
X_(115)Kiepert centerX_(110)focus of Kiepert parabola
X_(117)midpoint of X_4 and X_(109)X_(2988)Dc(X_(102))
X_(118)midpoint of X_4 and X_(101)X_(2989)Dc(X_(103))
X_(119)Feuerbach antipodeX_(2990)Dc(X_(104))
X_(120)X_(105)-of-medial triangleX_(2991)Dc(X_(105))
X_(125)Jerabek centerX_(648)trilinear pole of Euler line
X_(132)X_(105)-of-orthic triangleX_(287)X_2-Hirst inverse of X_(98)
X_(186)inverse-in-circumcircle of X_4X_(1994)orthocorrespondent of X_5
X_(242)X_4-line conjugate of X_(71)X_(1999)orthocorrespondent of X_(10)
X_(403)X_(36) of the orthic triangleX_(1993)orthocorrespondent of X_3
X_(468)X_2-line conjugate of X_3X_(1992)orthocorrespondent of X_2
X_(915)isogonal conjugate of X_(912)X_(2990)Dc(X_(104))
X_(917)X_(516)-cross conjugate of X_4X_(2989)Dc(X_(103))
X_(1300)Collings transform of X_(136)X_(2986)Dc(X_(74))
X_(1560)orthojoin of X_(468)X_(895)isogonal conjugate of X_(468)
X_(1566)orthojoin of X_(676)X_(677)trilinear pole of line X_3X_(101)
X_(1785)Mimosa transform of X_(1295)X_(57)isogonal conjugate of X_9
X_(1845)Zosma transform of X_(80)X_(2006)orthocorrespondent of X_(80)
X_(1878)Zosma transform of X_(519)X_(1997)orthocorrespondent of X_8

See also


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Gibert, B. "Orthocorrespondence and Orthopivotal Cubics." Forum Geom. 3, 1-27, 2003a., B. "Antiorthocorrespondents of Circumconics." Forum Geom. 3, 231-249, 2003b. B. "Orthopivotal Cubics.", B. and van Lamoen, F. M. "The Parasix Configuration and Orthocorrespondence." Forum Geom. 3, 169-180, 2003.

Referenced on Wolfram|Alpha


Cite this as:

Weisstein, Eric W. "Orthocorrespondent." From MathWorld--A Wolfram Web Resource.

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