Perspectrix

The line joining the three collinear points of intersection of the extensions of corresponding sides in perspective triangles, also called the perspective axis or homology axis.

The following table summarizes the perspectrices for various pairs of named triangles. Pairs in which one triangle is inscribed in another have the line at infinity as their perspectrix and are omitted in the list below.

triangle 1	triangle 2	perspectrix	Kimberling	line
anticomplementary triangle	circum-medial triangle	third power point		de Longchamps line
anticomplementary triangle	circum-orthic triangle	*	*	*
anticomplementary triangle	contact triangle	*	*	*
anticomplementary triangle	D-triangle	*	*	*
anticomplementary triangle	extouch triangle	*	*	*
anticomplementary triangle	first Brocard triangle	*	*	*
anticomplementary triangle	first Neuberg triangle	*	*	*
anticomplementary triangle	first Yff triangle	*	*	*
anticomplementary triangle	Fuhrmann triangle	*	*	*
anticomplementary triangle	incentral triangle	-Ceva conjugate of
anticomplementary triangle	inner Napoleon triangle	*	*	*
anticomplementary triangle	inner Vecten triangle	*	*	*
anticomplementary triangle	Lemoine triangle	*	*	*
anticomplementary triangle	MacBeath triangle	third power point		de Longchamps line
anticomplementary triangle	orthic triangle	*	*	*
anticomplementary triangle	outer Napoleon triangle	*	*	*
anticomplementary triangle	outer Vecten triangle	*	*	*
anticomplementary triangle	second Neuberg triangle	*	*	*
anticomplementary triangle	second Yff triangle	*	*	*
anticomplementary triangle	Steiner triangle	*	*	*
anticomplementary triangle	symmedial triangle	-Ceva conjugate of
anticomplementary triangle	third Brocard triangle	*	*	*
anticomplementary triangle	Yff contact triangle	exsimilicenter of Moses circle and incircle
BCI triangle	excentral triangle	*	*	*
BCI triangle	reference triangle	*	*	*
BCI triangle	tangential mid-arc triangle	*	*	*
circumcircle mid-arc triangle	first Brocard triangle	*	*	*
circumcircle mid-arc triangle	first Neuberg triangle	*	*	*
circumcircle mid-arc triangle	first Yff circles triangle	*	*	*
circumcircle mid-arc triangle	Fuhrmann triangle	psi(incenter, symmedian point)
circumcircle mid-arc triangle	incentral triangle	mittenpunkt
circumcircle mid-arc triangle	inner Napoleon triangle	*	*	*
circumcircle mid-arc triangle	inner Vecten triangle	*	*	*
circumcircle mid-arc triangle	Johnson triangle	*	*	*
circumcircle mid-arc triangle	medial triangle	-Ceva conjugate of
circumcircle mid-arc triangle	mid-arc triangle	*	*	*
circumcircle mid-arc triangle	mixtilinear triangle	crosspoint of and
circumcircle mid-arc triangle	outer Napoleon triangle	*	*	*
circumcircle mid-arc triangle	outer Vecten triangle	*	*	*
circumcircle mid-arc triangle	reference triangle	crosspoint of incenter and triangle centroid
circumcircle mid-arc triangle	second Neuberg triangle	*	*	*
circumcircle mid-arc triangle	second Yff circles triangle	*	*	*
circumcircle mid-arc triangle	tangential triangle	mittenpunkt
circum-medial triangle	D-triangle	*	*	*
circum-medial triangle	MacBeath triangle	third power point		de Longchamps line
circum-medial triangle	medial triangle	circumcenter		orthic axis
circum-medial triangle	reference triangle	Brocard midpoint
circum-medial triangle	tangential triangle	circumcenter		orthic axis
circum-orthic triangle	Euler triangle	circumcenter		orthic axis
circum-orthic triangle	half-altitude triangle	*	*	*
circum-orthic triangle	Johnson triangle	*	*	*
circum-orthic triangle	MacBeath triangle	harmonic of
circum-orthic triangle	reference triangle	-Ceva conjugate of
circum-orthic triangle	reflection triangle	*	*	*
contact triangle	first Yff circles triangle	*	*	*
contact triangle	half-altitude triangle	*	*	*
contact triangle	intangents triangle	-Ceva conjugate of
contact triangle	mid-arc triangle	isogonal conjugate of
contact triangle	reference triangle	internal similitude center of circumcircle and incircle		Gergonne line
contact triangle	second Yff circles triangle	*	*	*
contact triangle	tangential triangle	*	*	*
D-triangle	inner Napoleon triangle	focus of Kiepert parabola
D-triangle	medial triangle	tangential isogonal conjugate of
D-triangle	outer Napoleon triangle	focus of Kiepert parabola
D-triangle	reference triangle	isogonal conjugate of
Euler triangle	half-altitude triangle	*	*	*
Euler triangle	orthic triangle	harmonic of
Euler triangle	reflection triangle	*	*	*
excentral triangle	extangents triangle	incenter		antiorthic axis
excentral triangle	extouch triangle	-Ceva conjugate of
excentral triangle	Feuerbach triangle	incenter		antiorthic axis
excentral triangle	first Brocard triangle	*	*	*
excentral triangle	first Morley adjunct triangle	*	*	*
excentral triangle	first Morley triangle	*	*	*
excentral triangle	first Neuberg triangle	*	*	*
excentral triangle	first Yff circles triangle	*	*	*
excentral triangle	first Yff triangle	*	*	*
excentral triangle	Fuhrmann triangle	*	*	*
excentral triangle	half-altitude triangle	-Ceva conjugate of
excentral triangle	incentral triangle	incenter		antiorthic axis
excentral triangle	inner inscribed squares triangle	*	*	*
excentral triangle	inner Napoleon triangle	*	*	*
excentral triangle	inner Vecten triangle	*	*	*
excentral triangle	Lemoine triangle	*	*	*
excentral triangle	MacBeath triangle	*	*	*
excentral triangle	medial triangle	internal similitude center of circumcircle and incircle		Gergonne line
excentral triangle	mid-arc triangle	*	*	*
excentral triangle	mixtilinear triangle	*	*	*
excentral triangle	orthic triangle	*	*	*
excentral triangle	outer inscribed squares triangle	point Caroli II
excentral triangle	outer Napoleon triangle	*	*	*
excentral triangle	outer Vecten triangle	*	*	*
excentral triangle	reference triangle	incenter		antiorthic axis
excentral triangle	reflection triangle	*	*	*
excentral triangle	second Morley adjunct triangle	*	*	*
excentral triangle	second Morley triangle	*	*	*
excentral triangle	second Neuberg triangle	*	*	*
excentral triangle	second Yff circles triangle	*	*	*
excentral triangle	second Yff triangle	*	*	*
excentral triangle	Steiner triangle	exsimilicenter of Moses circle and incircle
excentral triangle	symmedial triangle	-Ceva conjugate of
excentral triangle	third Brocard triangle	*	*	*
excentral triangle	third Morley adjunct triangle	*	*	*
excentral triangle	third Morley triangle	*	*	*
excentral triangle	Yff contact triangle	*	*	*
extangents triangle	extouch triangle	-Ceva conjugate of
extangents triangle	Feuerbach triangle	incenter		antiorthic axis
extangents triangle	incentral triangle	incenter		antiorthic axis
extangents triangle	reference triangle	incenter		antiorthic axis
extouch triangle	half-altitude triangle	-Ceva conjugate of
extouch triangle	intangents triangle	*	*	*
extouch triangle	reference triangle	external similitude center of circumcircle and incircle
extouch triangle	tangential triangle	center of Yiu conic
Feuerbach triangle	incentral triangle	incenter		antiorthic axis
Feuerbach triangle	reference triangle	incenter		antiorthic axis
first Brocard triangle	first Neuberg triangle	isogonal conjugate of
first Brocard triangle	second Brocard triangle	*	*	*
first Brocard triangle	second Neuberg triangle	*	*	*
first Brocard triangle	third Brocard triangle	isogonal conjugate of
first Morley adjunct triangle	first Morley triangle	*	*	*
first Neuberg triangle	second Neuberg triangle	*	*	*
Fuhrmann triangle	first Brocard triangle	*	*	*
Fuhrmann triangle	first Neuberg triangle	*	*	*
Fuhrmann triangle	inner Napoleon triangle	*	*	*
Fuhrmann triangle	inner Vecten triangle	*	*	*
Fuhrmann triangle	Johnson triangle	*	*	*
Fuhrmann triangle	medial triangle	*	*	*
Fuhrmann triangle	outer Napoleon triangle	*	*	*
Fuhrmann triangle	outer Vecten triangle	*	*	*
Fuhrmann triangle	second Neuberg triangle	*	*	*
Fuhrmann triangle	tangential triangle	*	*	*
half-altitude triangle	inner inscribed squares triangle	isogonal conjugate of
half-altitude triangle	medial triangle	-Ceva conjugate of
half-altitude triangle	orthic triangle	reflection of in
half-altitude triangle	outer inscribed squares triangle	isogonal conjugate of
half-altitude triangle	reference triangle	isogonal conjugate of
half-altitude triangle	reflection triangle	*	*	*
hexyl triangle	first Yff circles triangle	*	*	*
hexyl triangle	intangents triangle	mittenpunkt
hexyl triangle	reference triangle	Clawson point
hexyl triangle	second Yff circles triangle	*	*	*
incentral triangle	first Yff circles triangle	*	*	*
incentral triangle	intangents triangle	Mimosa transform of
incentral triangle	mid-arc triangle	*	*	*
incentral triangle	mixtilinear triangle	*	*	*
incentral triangle	reference triangle	incenter		antiorthic axis
incentral triangle	second Yff circles triangle	*	*	*
incentral triangle	tangential triangle	mittenpunkt
inner inscribed squares triangle	outer inscribed squares triangle	circumcenter		orthic axis
inner Napoleon triangle	first Brocard triangle	*	*	*
inner Napoleon triangle	first Neuberg triangle	*	*	*
inner Napoleon triangle	inner Vecten triangle	*	*	*
inner Napoleon triangle	Johnson triangle	*	*	*
inner Napoleon triangle	medial triangle	*	*	*
inner Napoleon triangle	outer Napoleon triangle	focus of Kiepert parabola
inner Napoleon triangle	outer Vecten triangle	*	*	*
inner Napoleon triangle	reference triangle	*	*	*
inner Napoleon triangle	second Neuberg triangle	*	*	*
inner Napoleon triangle	tangential triangle	*	*	*
inner Vecten triangle	first Brocard triangle	*	*	*
inner Vecten triangle	first Neuberg triangle	*	*	*
inner Vecten triangle	inner inscribed squares triangle	*	*	*
inner Vecten triangle	Johnson triangle	*	*	*
inner Vecten triangle	medial triangle	*	*	*
inner Vecten triangle	orthic triangle	*	*	*
inner Vecten triangle	outer inscribed squares triangle	*	*	*
inner Vecten triangle	outer Napoleon triangle	*	*	*
inner Vecten triangle	outer Vecten triangle	*	*	*
inner Vecten triangle	reference triangle	second van Lamoen homothetic center
inner Vecten triangle	second Neuberg triangle	*	*	*
inner Vecten triangle	tangential triangle	*	*	*
Johnson triangle	first Brocard triangle	*	*	*
Johnson triangle	first Neuberg triangle	*	*	*
Johnson triangle	MacBeath triangle	orthoharmonic of
Johnson triangle	orthic triangle	*	*	*
Johnson triangle	outer Napoleon triangle	*	*	*
Johnson triangle	outer Vecten triangle	*	*	*
Johnson triangle	second Neuberg triangle	*	*	*
Johnson triangle	tangential triangle	harmonic of
Johnson triangle	Yiu triangle	*	*	*
Lemoine triangle	reference triangle	harmonic of
Lemoine triangle	tangential triangle	*	*	*
Lucas central triangle	first Neuberg triangle	*	*	*
Lucas central triangle	Lucas inner triangle	triangle centroid		Lemoine axis
Lucas central triangle	Lucas tangents triangle	triangle centroid		Lemoine axis
Lucas central triangle	reference triangle	triangle centroid		Lemoine axis
Lucas central triangle	symmedial triangle	triangle centroid		Lemoine axis
Lucas central triangle	tangential triangle	triangle centroid		Lemoine axis
Lucas inner triangle	Lucas tangents triangle	triangle centroid		Lemoine axis
Lucas inner triangle	reference triangle	triangle centroid		Lemoine axis
Lucas inner triangle	symmedial triangle	triangle centroid		Lemoine axis
Lucas inner triangle	tangential triangle	triangle centroid		Lemoine axis
Lucas tangents triangle	reference triangle	triangle centroid		Lemoine axis
Lucas tangents triangle	symmedial triangle	triangle centroid		Lemoine axis
Lucas tangents triangle	tangential triangle	triangle centroid		Lemoine axis
MacBeath triangle	reference triangle	inverse of in the Brocard circle
MacBeath triangle	tangential triangle	*	*	*
medial triangle	first Brocard triangle	*	*	*
medial triangle	first Neuberg triangle	TCC-perspector of
medial triangle	inner inscribed squares triangle	*	*	*
medial triangle	mid-arc triangle	*	*	*
medial triangle	outer inscribed squares triangle	*	*	*
medial triangle	outer Napoleon triangle	*	*	*
medial triangle	outer Vecten triangle	*	*	*
medial triangle	second Neuberg triangle	*	*	*
medial triangle	tangential triangle	circumcenter		orthic axis
mid-arc triangle	first Yff circles triangle	*	*	*
mid-arc triangle	mixtilinear triangle	*	*	*
mid-arc triangle	reference triangle	*	*	*
mid-arc triangle	second Yff circles triangle	*	*	*
mid-arc triangle	tangential mid-arc triangle	*	*	*
mixtilinear triangle	first Yff circles triangle	*	*	*
mixtilinear triangle	reference triangle	mittenpunkt
mixtilinear triangle	second Yff circles triangle	*	*	*
orthic triangle	inner inscribed squares triangle	circumcenter		orthic axis
orthic triangle	outer inscribed squares triangle	circumcenter		orthic axis
orthic triangle	outer Vecten triangle	*	*	*
orthic triangle	reference triangle	circumcenter		orthic axis
orthic triangle	reflection triangle	-Ceva conjugate of
outer Napoleon triangle	first Brocard triangle	*	*	*
outer Napoleon triangle	first Neuberg triangle	*	*	*
outer Napoleon triangle	outer Vecten triangle	*	*	*
outer Napoleon triangle	reference triangle	DC()
outer Napoleon triangle	second Neuberg triangle	*	*	*
outer Napoleon triangle	tangential triangle	*	*	*
outer Vecten triangle	first Brocard triangle	*	*	*
outer Vecten triangle	first Neuberg triangle	*	*	*
outer Vecten triangle	inner inscribed squares triangle	*	*	*
outer Vecten triangle	outer inscribed squares triangle	*	*	*
outer Vecten triangle	reference triangle	first van Lamoen homothetic center
outer Vecten triangle	second Neuberg triangle	*	*	*
outer Vecten triangle	tangential triangle	*	*	*
reference triangle	first Brocard triangle	isogonal conjugate of
reference triangle	first Morley adjunct triangle	*	*	*
reference triangle	first Morley triangle	*	*	*
reference triangle	first Neuberg triangle	isogonal conjugate of
reference triangle	first Yff triangle	*	*	*
reference triangle	inner inscribed squares triangle	circumcenter		orthic axis
reference triangle	outer inscribed squares triangle	circumcenter		orthic axis
reference triangle	reflection triangle	Kosnita point
reference triangle	second Brocard triangle	Parry point
reference triangle	second Morley adjunct triangle	*	*	*
reference triangle	second Morley triangle	*	*	*
reference triangle	second Neuberg triangle	*	*	*
reference triangle	second Yff triangle	*	*	*
reference triangle	Steiner triangle	isogonal conjugate of
reference triangle	symmedial triangle	triangle centroid		Lemoine axis
reference triangle	tangential mid-arc triangle	isogonal conjugate of
reference triangle	tangential triangle	triangle centroid		Lemoine axis
reference triangle	third Brocard triangle	isogonal conjugate of
reference triangle	third Morley adjunct triangle	*	*	*
reference triangle	third Morley triangle	*	*	*
reference triangle	Yff central triangle	*	*	*
reference triangle	Yff contact triangle	crossdifference of and		Nagel line
reference triangle	Yiu triangle	barycentric product
second Morley adjunct triangle	second Morley triangle	*	*	*
Steiner triangle	tangential triangle	crosspoint of and
symmedial triangle	second Brocard triangle	*	*	*
symmedial triangle	tangential triangle	triangle centroid		Lemoine axis
symmedial triangle	third Brocard triangle	*	*	*
tangential triangle	first Brocard triangle	*	*	*
tangential triangle	first Neuberg triangle	*	*	*
tangential triangle	first Yff triangle	*	*	*
tangential triangle	second Brocard triangle	*	*	*
tangential triangle	second Neuberg triangle	*	*	*
tangential triangle	second Yff triangle	*	*	*
tangential triangle	third Brocard triangle	*	*	*
tangential triangle	Yff contact triangle	*	*	*
tangential triangle	Yiu triangle	*	*	*
third Brocard triangle	first Neuberg triangle	isogonal conjugate of
third Morley adjunct triangle	third Morley triangle	*	*	*