Perspector

The point at which the three lines connecting the vertices of two perspective triangles concur, sometimes also called the perspective center, homology center, or pole.

In the plane of a reference triangle , the perspector of and its polar triangle with respect to a given conic is called the perspector of that conic. The perspector is not defined for conics with respect to which is self-polar. For an inconic, the perspector is the Brianchon point of the conic.

The following table summarizes perspectors of some named triangle conics corresponding to Kimberling centers.

conic	Kimberling	perspector
anticomplementary triangle		orthocenter
Bevan circle		isogonal conjugate of
Brocard inellipse		symmedian point
circumcircle		symmedian point
de Longchamps circle		-cross conjugate of
excircles radical circle		exsimilicenter(nine-point circle, Apollonius circle)
Feuerbach hyperbola		crossdifference of and
first Droz-Farny circle		orthocenter
incircle		Gergonne point
Jerabek hyperbola		crossdifference of and
Kiepert hyperbola		isogonal conjugate of
Kiepert parabola		Steiner point
Lemoine inellipse		isogonal conjugate of
Macbeath circumconic		circumcenter
Macbeath inconic		isotomic conjugate of the circumcenter
Mandart inellipse		Nagel point
orthic inconic		orthocenter
Spieker circle		triangle centroid
Steiner circumellipse		triangle centroid
Steiner inellipse		triangle centroid
tangential circle
tangential mid-arc circle		third mid-arc point
Yff parabola		Yff parabolic point

The following table lists the perspectors of pairs of special triangles.

first triangle	second triangle	perspector	Kimberling
anticomplementary triangle	circum-medial triangle	triangle centroid
anticomplementary triangle	circum-orthic triangle	*	*
anticomplementary triangle	contact triangle	anticomplement of Nagel point
anticomplementary triangle	D-triangle	triangle centroid
anticomplementary triangle	Euler triangle	*	*
anticomplementary triangle	extouch triangle	anticomplement of
anticomplementary triangle	first Brocard triangle	anticomplementary conjugate of
anticomplementary triangle	first Neuberg triangle	*	*
anticomplementary triangle	first Yff circles triangle	*	*
anticomplementary triangle	first Yff triangle	*	*
anticomplementary triangle	Fuhrmann triangle	Nagel point
anticomplementary triangle	incentral triangle	-Ceva conjugate of
anticomplementary triangle	inner Napoleon triangle	anticomplement of
anticomplementary triangle	inner Vecten triangle	anticomplement of
anticomplementary triangle	Johnson triangle	orthocenter
anticomplementary triangle	Lemoine triangle	orthocorrespondent of
anticomplementary triangle	Macbeath triangle	orthocenter
anticomplementary triangle	medial triangle	triangle centroid
anticomplementary triangle	orthic triangle	-Ceva conjugate of
anticomplementary triangle	outer Napoleon triangle	anticomplement of
anticomplementary triangle	outer Vecten triangle	anticomplement of
anticomplementary triangle	reference triangle	triangle centroid
anticomplementary triangle	second Neuberg triangle	Tarry point of anticomplementary triangle
anticomplementary triangle	second Yff circles triangle	*	*
anticomplementary triangle	second Yff triangle	*	*
anticomplementary triangle	Steiner triangle	isogonal conjugate of
anticomplementary triangle	symmedial triangle	-Ceva conjugate of
anticomplementary triangle	third Brocard triangle	-Ceva conjugate of
anticomplementary triangle	Yff contact triangle	isogonal conjugate of
BCI triangle	excentral triangle	*	*
BCI triangle	reference triangle	first de Villiers point
BCI triangle	tangential mid-arc triangle	incenter
circumcircle mid-arc triangle	contact triangle	external similitude center of circumcircle and incircle
circumcircle mid-arc triangle	excentral triangle	incenter
circumcircle mid-arc triangle	first Brocard triangle	circumcenter
circumcircle mid-arc triangle	first Neuberg triangle	circumcenter
circumcircle mid-arc triangle	first Yff circles triangle	incenter
circumcircle mid-arc triangle	Fuhrmann triangle	circumcenter
circumcircle mid-arc triangle	hexyl triangle	*	*
circumcircle mid-arc triangle	incentral triangle	incenter
circumcircle mid-arc triangle	inner Napoleon triangle	circumcenter
circumcircle mid-arc triangle	inner Vecten triangle	circumcenter
circumcircle mid-arc triangle	Johnson triangle	circumcenter
circumcircle mid-arc triangle	medial triangle	circumcenter
circumcircle mid-arc triangle	mid-arc triangle	incenter
circumcircle mid-arc triangle	mixtilinear triangle	incenter
circumcircle mid-arc triangle	outer Napoleon triangle	circumcenter
circumcircle mid-arc triangle	outer Vecten triangle	circumcenter
circumcircle mid-arc triangle	reference triangle	incenter
circumcircle mid-arc triangle	second Neuberg triangle	circumcenter
circumcircle mid-arc triangle	second Yff circles triangle	incenter
circumcircle mid-arc triangle	tangential mid-arc triangle	*	*
circumcircle mid-arc triangle	tangential triangle	circumcenter
circumcircle mid-arc triangle	Yff central triangle	*	*
circum-medial triangle	D-triangle	triangle centroid
circum-medial triangle	Macbeath triangle	homothetic center of orthic and tangential triangles
circum-medial triangle	medial triangle	triangle centroid
circum-medial triangle	reference triangle	triangle centroid
circum-medial triangle	tangential triangle	Exeter point
circumnormal triangle	circumtangential triangle	circumcenter
circumnormal triangle	first Morley triangle	*	*
circumnormal triangle	second Morley triangle	*	*
circumnormal triangle	Stammler triangle	circumcenter
circumnormal triangle	third Morley triangle	*	*
circum-orthic triangle	Euler triangle	orthocenter
circum-orthic triangle	extangents triangle	*	*
circum-orthic triangle	half-altitude triangle	orthocenter
circum-orthic triangle	intangents triangle	*	*
circum-orthic triangle	Johnson triangle	focus of Kiepert parabola
circum-orthic triangle	Macbeath triangle	circumcenter
circum-orthic triangle	orthic triangle	orthocenter
circum-orthic triangle	reference triangle	orthocenter
circum-orthic triangle	reflection triangle	orthocenter
circum-orthic triangle	tangential triangle	perspector of abc and orthic-of-orthic triangle
circumtangential triangle	first Morley triangle	*	*
circumtangential triangle	second Morley triangle	*	*
circumtangential triangle	Stammler triangle	circumcenter
circumtangential triangle	third Morley triangle	*	*
contact triangle	excentral triangle	isogonal conjugate of
contact triangle	first Yff circles triangle	*	*
contact triangle	half-altitude triangle	isogonal conjugate of
contact triangle	hexyl triangle	incenter
contact triangle	intangents triangle	incenter
contact triangle	mid-arc triangle	first mid-arc point
contact triangle	reference triangle	Gergonne point
contact triangle	second Yff circles triangle	*	*
contact triangle	tangential mid-arc triangle	third mid-arc point
contact triangle	tangential triangle	perspector contact and tangential triangles
contact triangle	Yff central triangle	Yff center of congruence
D-triangle	inner Napoleon triangle	second isodynamic point
D-triangle	medial triangle	triangle centroid
D-triangle	outer Napoleon triangle	first isodynamic point
D-triangle	reference triangle	triangle centroid
Euler triangle	first Yff circles triangle	(,)-harmonic conjugate of
Euler triangle	half-altitude triangle	orthocenter
Euler triangle	Johnson triangle	midpoint of and
Euler triangle	medial triangle	nine-point center
Euler triangle	orthic triangle	orthocenter
Euler triangle	reference triangle	orthocenter
Euler triangle	reflection triangle	orthocenter
Euler triangle	second Yff circles triangle	center of first Johnson-Yff circle
excentral triangle	extangents triangle	Bevan point
excentral triangle	extouch triangle	Bevan point
excentral triangle	Feuerbach triangle	nine-point center
excentral triangle	first Brocard triangle	*	*
excentral triangle	first Morley adjunct triangle	second Morley-Gibert point
excentral triangle	first Morley triangle	first Morley-Gibert point
excentral triangle	first Neuberg triangle	*	*
excentral triangle	first Yff circles triangle	incenter
excentral triangle	first Yff triangle	*	*
excentral triangle	Fuhrmann triangle	-Ceva conjugate of
excentral triangle	half-altitude triangle	isogonal conjugate of
excentral triangle	hexyl triangle	circumcenter
excentral triangle	incentral triangle	incenter
excentral triangle	inner inscribed squares triangle	*	*
excentral triangle	inner Napoleon triangle	*	*
excentral triangle	inner Vecten triangle	*	*
excentral triangle	Lemoine triangle	*	*
excentral triangle	Macbeath triangle	Mimosa transform of
excentral triangle	medial triangle	mittenpunkt
excentral triangle	mid-arc triangle	incenter
excentral triangle	mixtilinear triangle	incenter
excentral triangle	orthic triangle	-Ceva conjugate of
excentral triangle	outer inscribed squares triangle	*	*
excentral triangle	outer Napoleon triangle	*	*
excentral triangle	outer Vecten triangle	*	*
excentral triangle	reference triangle	incenter
excentral triangle	reflection triangle	first Evans perspector
excentral triangle	second Morley adjunct triangle	*	*
excentral triangle	second Morley triangle	*	*
excentral triangle	second Neuberg triangle	*	*
excentral triangle	second Yff circles triangle	incenter
excentral triangle	second Yff triangle	*	*
excentral triangle	Steiner triangle	-Ceva conjugate of
excentral triangle	symmedial triangle	-Ceva conjugate of
excentral triangle	tangential mid-arc triangle	*	*
excentral triangle	third Brocard triangle	*	*
excentral triangle	third Morley adjunct triangle	*	*
excentral triangle	third Morley triangle	*	*
excentral triangle	Yff central triangle	congruent isoscelizers point
excentral triangle	Yff contact triangle	crossdifference of and
extangents triangle	extouch triangle	Bevan point
extangents triangle	Feuerbach triangle	*	*
extangents triangle	incentral triangle	internal similitude center of circumcircle and incircle
extangents triangle	intangents triangle	internal similitude center of circumcircle and incircle
extangents triangle	orthic triangle	Clawson point
extangents triangle	reference triangle	orthocenter of the contact triangle
extangents triangle	tangential triangle	internal similitude center of circumcircle and incircle
extouch triangle	half-altitude triangle	incenter
extouch triangle	intangents triangle	-Ceva conjugate of
extouch triangle	reference triangle	Nagel point
extouch triangle	tangential triangle	-Ceva conjugate of
Feuerbach triangle	incentral triangle	Feuerbach point
Feuerbach triangle	reference triangle	(,)-harmonic conjugate of
first Brocard triangle	first Neuberg triangle	circumcenter
first Brocard triangle	second Brocard triangle	triangle centroid
first Brocard triangle	second Neuberg triangle	circumcenter
first Brocard triangle	third Brocard triangle	Euler line intercept of line
first Morley adjunct triangle	first Morley triangle	first Morley center
first Morley triangle	second Morley triangle	second Morley-Taylor-Marr center
first Morley triangle	third Morley triangle	4th Morley-Taylor-Marr center
first Neuberg triangle	second Neuberg triangle	circumcenter
first Yff circles triangle	second Yff circles triangle	incenter
Fuhrmann triangle	first Brocard triangle	circumcenter
Fuhrmann triangle	first Neuberg triangle	circumcenter
Fuhrmann triangle	inner Napoleon triangle	circumcenter
Fuhrmann triangle	inner Vecten triangle	circumcenter
Fuhrmann triangle	Johnson triangle	circumcenter
Fuhrmann triangle	medial triangle	circumcenter
Fuhrmann triangle	outer Napoleon triangle	circumcenter
Fuhrmann triangle	outer Vecten triangle	circumcenter
Fuhrmann triangle	second Neuberg triangle	circumcenter
Fuhrmann triangle	tangential triangle	circumcenter
half-altitude triangle	inner inscribed squares triangle	symmedian point
half-altitude triangle	medial triangle	symmedian point
half-altitude triangle	orthic triangle	orthocenter
half-altitude triangle	outer inscribed squares triangle	symmedian point
half-altitude triangle	reference triangle	orthocenter
half-altitude triangle	reflection triangle	orthocenter
hexyl triangle	first Yff circles triangle	*	*
hexyl triangle	intangents triangle	incenter
hexyl triangle	reference triangle	isogonal conjugate of
hexyl triangle	second Yff circles triangle	*	*
hexyl triangle	tangential mid-arc triangle	*	*
hexyl triangle	Yff central triangle	*	*
incentral triangle	first Yff circles triangle	incenter
incentral triangle	intangents triangle	internal similitude center of circumcircle and incircle
incentral triangle	mid-arc triangle	incenter
incentral triangle	mixtilinear triangle	incenter
incentral triangle	reference triangle	incenter
incentral triangle	second Yff circles triangle	incenter
incentral triangle	tangential triangle	internal similitude center of circumcircle and incircle
inner inscribed squares triangle	outer inscribed squares triangle	symmedian point
inner Napoleon triangle	first Brocard triangle	circumcenter
inner Napoleon triangle	first Neuberg triangle	circumcenter
inner Napoleon triangle	inner Vecten triangle	circumcenter
inner Napoleon triangle	Johnson triangle	circumcenter
inner Napoleon triangle	medial triangle	circumcenter
inner Napoleon triangle	outer Napoleon triangle	circumcenter
inner Napoleon triangle	outer Vecten triangle	circumcenter
inner Napoleon triangle	reference triangle	second Napoleon point
inner Napoleon triangle	second Neuberg triangle	circumcenter
inner Napoleon triangle	tangential triangle	circumcenter
inner Vecten triangle	first Brocard triangle	circumcenter
inner Vecten triangle	first Neuberg triangle	circumcenter
inner Vecten triangle	inner inscribed squares triangle	Kenmotu point
inner Vecten triangle	Johnson triangle	circumcenter
inner Vecten triangle	medial triangle	circumcenter
inner Vecten triangle	orthic triangle	Kenmotu point
inner Vecten triangle	outer inscribed squares triangle	inner Vecten point
inner Vecten triangle	outer Napoleon triangle	circumcenter
inner Vecten triangle	outer Vecten triangle	circumcenter
inner Vecten triangle	reference triangle	inner Vecten point
inner Vecten triangle	second Neuberg triangle	circumcenter
inner Vecten triangle	tangential triangle	circumcenter
intangents triangle	orthic triangle	perspector of the orthic and intangents triangles
intangents triangle	tangential triangle	internal similitude center of circumcircle and incircle
Johnson triangle	first Brocard triangle	circumcenter
Johnson triangle	first Neuberg triangle	circumcenter
Johnson triangle	first Yff circles triangle	(,)-harmonic conjugate of
Johnson triangle	Macbeath triangle	orthocenter
Johnson triangle	medial triangle	circumcenter
Johnson triangle	orthic triangle	eigencenter of orthic triangle
Johnson triangle	outer Napoleon triangle	circumcenter
Johnson triangle	outer Vecten triangle	circumcenter
Johnson triangle	reference triangle	nine-point center
Johnson triangle	second Neuberg triangle	circumcenter
Johnson triangle	second Yff circles triangle	Feuerbach point
Johnson triangle	tangential triangle	circumcenter
Johnson triangle	Yiu triangle	nine-point center
Lemoine triangle	reference triangle	isogonal conjugate of
Lemoine triangle	tangential triangle	orthocorrespondent of
Lucas central triangle	first Neuberg triangle	*	*
Lucas central triangle	Lucas inner triangle	*	*
Lucas central triangle	Lucas tangents triangle	*	*
Lucas central triangle	reference triangle	circumcenter
Lucas central triangle	symmedial triangle	isogonal conjugate of
Lucas central triangle	tangential triangle	Kenmotu point
Lucas inner triangle	Lucas tangents triangle	*	*
Lucas inner triangle	reference triangle	isogonal conjugate of
Lucas inner triangle	symmedial triangle	*	*
Lucas inner triangle	tangential triangle	*	*
Lucas tangents triangle	reference triangle	Kenmotu point
Lucas tangents triangle	symmedial triangle	circumcenter
Lucas tangents triangle	tangential triangle	isogonal conjugate of
Macbeath triangle	reference triangle	isotomic conjugate of circumcenter
Macbeath triangle	tangential triangle	-of-tangential triangle
medial triangle	first Brocard triangle	circumcenter
medial triangle	first Neuberg triangle	circumcenter
medial triangle	first Yff circles triangle	Yff concurrent congruent circles point
medial triangle	inner inscribed squares triangle	symmedian point
medial triangle	mid-arc triangle	second mid-arc point
medial triangle	outer inscribed squares triangle	symmedian point
medial triangle	outer Napoleon triangle	circumcenter
medial triangle	outer Vecten triangle	circumcenter
medial triangle	reference triangle	triangle centroid
medial triangle	second Neuberg triangle	circumcenter
medial triangle	second Yff circles triangle	(,)-harmonic conjugate of
medial triangle	tangential triangle	circumcenter
mid-arc triangle	first Yff circles triangle	incenter
mid-arc triangle	mixtilinear triangle	incenter
mid-arc triangle	reference triangle	incenter
mid-arc triangle	second Yff circles triangle	incenter
mid-arc triangle	tangential mid-arc triangle	*	*
mixtilinear triangle	first Yff circles triangle	incenter
mixtilinear triangle	reference triangle	incenter
mixtilinear triangle	second Yff circles triangle	incenter
orthic triangle	inner inscribed squares triangle	Kenmotu point
orthic triangle	outer inscribed squares triangle	(,)-harmonic conjugate of
orthic triangle	outer Vecten triangle	(,)-harmonic conjugate of
orthic triangle	reference triangle	orthocenter
orthic triangle	reflection triangle	orthocenter
orthic triangle	tangential triangle	homothetic center of orthic and tangential triangles
outer Napoleon triangle	first Brocard triangle	circumcenter
outer Napoleon triangle	first Neuberg triangle	circumcenter
outer Napoleon triangle	outer Vecten triangle	circumcenter
outer Napoleon triangle	reference triangle	first Napoleon point
outer Napoleon triangle	second Neuberg triangle	circumcenter
outer Napoleon triangle	tangential triangle	circumcenter
outer Vecten triangle	first Brocard triangle	circumcenter
outer Vecten triangle	first Neuberg triangle	circumcenter
outer Vecten triangle	inner inscribed squares triangle	outer Vecten point
outer Vecten triangle	outer inscribed squares triangle	(,)-harmonic conjugate of
outer Vecten triangle	reference triangle	outer Vecten point
outer Vecten triangle	second Neuberg triangle	circumcenter
outer Vecten triangle	tangential triangle	circumcenter
reference triangle	first Brocard triangle	third Brocard point
reference triangle	first Morley adjunct triangle	second Morley-Taylor-Marr center
reference triangle	first Morley triangle	first Morley-Taylor-Marr center
reference triangle	first Neuberg triangle	Tarry point
reference triangle	first Yff circles triangle	incenter
reference triangle	first Yff triangle	*	*
reference triangle	inner inscribed squares triangle	outer Vecten point
reference triangle	outer inscribed squares triangle	inner Vecten point
reference triangle	reflection triangle	orthocenter
reference triangle	second Brocard triangle	symmedian point
reference triangle	second Morley adjunct triangle	6th Morley-Taylor-Marr center
reference triangle	second Morley triangle	5th Morley-Taylor-Marr center
reference triangle	second Neuberg triangle	isogonal conjugate of
reference triangle	second Yff circles triangle	incenter
reference triangle	second Yff triangle	*	*
reference triangle	Steiner triangle	Steiner point
reference triangle	symmedial triangle	symmedian point
reference triangle	tangential mid-arc triangle	first mid-arc point
reference triangle	tangential triangle	symmedian point
reference triangle	third Brocard triangle	third power point
reference triangle	third Morley adjunct triangle	4th Morley-Taylor-Marr center
reference triangle	third Morley triangle	third Morley-Taylor-Marr center
reference triangle	Yff central triangle	first mid-arc point
reference triangle	Yff contact triangle	Yff parabolic point
reference triangle	Yiu triangle	nine-point center
second Morley adjunct triangle	second Morley triangle	*	*
second Morley triangle	third Morley triangle	6th Morley-Taylor-Marr center
Stammler triangle	first Morley triangle	*	*
Stammler triangle	second Morley triangle	*	*
Stammler triangle	third Morley triangle	*	*
Steiner triangle	tangential triangle	crossdifference of and
symmedial triangle	second Brocard triangle	symmedian point
symmedial triangle	tangential triangle	symmedian point
symmedial triangle	third Brocard triangle	-Ceva conjugate of
tangential mid-arc triangle	Yff central triangle	first mid-arc point
tangential triangle	first Brocard triangle	circumcenter
tangential triangle	first Neuberg triangle	circumcenter
tangential triangle	first Yff triangle	*	*
tangential triangle	second Brocard triangle	symmedian point
tangential triangle	second Neuberg triangle	circumcenter
tangential triangle	second Yff triangle	*	*
tangential triangle	third Brocard triangle	*	*
tangential triangle	Yff contact triangle	*	*
tangential triangle	Yiu triangle	-Ceva conjugate of
third Brocard triangle	first Neuberg triangle	orthocenter
third Morley adjunct triangle	third Morley triangle	*	*