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Perspector


PerspectiveTriangles

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 DeltaABC, the perspector of DeltaABC 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 DeltaABC 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.

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

first trianglesecond triangleperspectorKimberling
anticomplementary trianglecircum-medial triangletriangle centroidX_2
anticomplementary trianglecircum-orthic triangle**
anticomplementary trianglecontact triangleanticomplement of Nagel pointX_(145)
anticomplementary triangleD-triangletriangle centroidX_2
anticomplementary triangleEuler triangle**
anticomplementary triangleextouch triangleanticomplement of X_7X_(144)
anticomplementary trianglefirst Brocard triangleanticomplementary conjugate of X_(76)X_(2896)
anticomplementary trianglefirst Neuberg triangle**
anticomplementary trianglefirst Yff circles triangle**
anticomplementary trianglefirst Yff triangle**
anticomplementary triangleFuhrmann triangleNagel pointX_8
anticomplementary triangleincentral triangleX_1-Ceva conjugate of X_2X_(192)
anticomplementary triangleinner Napoleon triangleanticomplement of X_(17)X_(627)
anticomplementary triangleinner Vecten triangleanticomplement of X_(485)X_(488)
anticomplementary triangleJohnson triangleorthocenterX_4
anticomplementary triangleLemoine triangleorthocorrespondent of X_2X_(1992)
anticomplementary triangleMacBeath triangleorthocenterX_4
anticomplementary trianglemedial triangletriangle centroidX_2
anticomplementary triangleorthic triangleX_4-Ceva conjugate of X_2X_(193)
anticomplementary triangleouter Napoleon triangleanticomplement of X_(18)X_(628)
anticomplementary triangleouter Vecten triangleanticomplement of X_(486)X_(487)
anticomplementary trianglereference triangletriangle centroidX_2
anticomplementary trianglesecond Neuberg triangleTarry point of anticomplementary triangleX_(147)
anticomplementary trianglesecond Yff circles triangle**
anticomplementary trianglesecond Yff triangle**
anticomplementary triangleSteiner triangleisogonal conjugate of X_(110)X_(523)
anticomplementary trianglesymmedial triangleX_6-Ceva conjugate of X_2X_(194)
anticomplementary trianglethird Brocard triangleX_6-Ceva conjugate of X_2X_(194)
anticomplementary triangleYff contact triangleisogonal conjugate of X_(101)X_(514)
BCI triangleexcentral triangle**
BCI trianglereference trianglefirst de Villiers pointX_(1127)
BCI triangletangential mid-arc triangleincenterX_1
circumcircle mid-arc trianglecontact triangleexternal similitude center of circumcircle and incircleX_(56)
circumcircle mid-arc triangleexcentral triangleincenterX_1
circumcircle mid-arc trianglefirst Brocard trianglecircumcenterX_3
circumcircle mid-arc trianglefirst Neuberg trianglecircumcenterX_3
circumcircle mid-arc trianglefirst Yff circles triangleincenterX_1
circumcircle mid-arc triangleFuhrmann trianglecircumcenterX_3
circumcircle mid-arc trianglehexyl triangle**
circumcircle mid-arc triangleincentral triangleincenterX_1
circumcircle mid-arc triangleinner Napoleon trianglecircumcenterX_3
circumcircle mid-arc triangleinner Vecten trianglecircumcenterX_3
circumcircle mid-arc triangleJohnson trianglecircumcenterX_3
circumcircle mid-arc trianglemedial trianglecircumcenterX_3
circumcircle mid-arc trianglemid-arc triangleincenterX_1
circumcircle mid-arc trianglemixtilinear triangleincenterX_1
circumcircle mid-arc triangleouter Napoleon trianglecircumcenterX_3
circumcircle mid-arc triangleouter Vecten trianglecircumcenterX_3
circumcircle mid-arc trianglereference triangleincenterX_1
circumcircle mid-arc trianglesecond Neuberg trianglecircumcenterX_3
circumcircle mid-arc trianglesecond Yff circles triangleincenterX_1
circumcircle mid-arc triangletangential mid-arc triangle**
circumcircle mid-arc triangletangential trianglecircumcenterX_3
circumcircle mid-arc triangleYff central triangle**
circum-medial triangleD-triangletriangle centroidX_2
circum-medial triangleMacBeath trianglehomothetic center of orthic and tangential trianglesX_(25)
circum-medial trianglemedial triangletriangle centroidX_2
circum-medial trianglereference triangletriangle centroidX_2
circum-medial triangletangential triangleExeter pointX_(22)
circumnormal trianglecircumtangential trianglecircumcenterX_3
circumnormal trianglefirst Morley triangle**
circumnormal trianglesecond Morley triangle**
circumnormal triangleStammler trianglecircumcenterX_3
circumnormal trianglethird Morley triangle**
circum-orthic triangleEuler triangleorthocenterX_4
circum-orthic triangleextangents triangle**
circum-orthic trianglehalf-altitude triangleorthocenterX_4
circum-orthic triangleintangents triangle**
circum-orthic triangleJohnson trianglefocus of Kiepert parabolaX_(110)
circum-orthic triangleMacBeath trianglecircumcenterX_3
circum-orthic triangleorthic triangleorthocenterX_4
circum-orthic trianglereference triangleorthocenterX_4
circum-orthic trianglereflection triangleorthocenterX_4
circum-orthic triangletangential triangleperspector of abc and orthic-of-orthic triangleX_(24)
circumtangential trianglefirst Morley triangle**
circumtangential trianglesecond Morley triangle**
circumtangential triangleStammler trianglecircumcenterX_3
circumtangential trianglethird Morley triangle**
contact triangleexcentral triangleisogonal conjugate of X_9X_(57)
contact trianglefirst Yff circles triangle**
contact trianglehalf-altitude triangleisogonal conjugate of X_9X_(57)
contact trianglehexyl triangleincenterX_1
contact triangleintangents triangleincenterX_1
contact trianglemid-arc trianglefirst mid-arc pointX_(177)
contact trianglereference triangleGergonne pointX_7
contact trianglesecond Yff circles triangle**
contact triangletangential mid-arc trianglethird mid-arc pointX_(2089)
contact triangletangential triangleperspector contact and tangential trianglesX_(1486)
contact triangleYff central triangleYff center of congruenceX_(174)
D-triangleinner Napoleon trianglesecond isodynamic pointX_(16)
D-trianglemedial triangletriangle centroidX_2
D-triangleouter Napoleon trianglefirst isodynamic pointX_(15)
D-trianglereference triangletriangle centroidX_2
Euler trianglefirst Yff circles triangle(X_1,X_4)-harmonic conjugate of X_(1478)X_(1479)
Euler trianglehalf-altitude triangleorthocenterX_4
Euler triangleJohnson trianglemidpoint of X_2 and X_4X_(381)
Euler trianglemedial trianglenine-point centerX_5
Euler triangleorthic triangleorthocenterX_4
Euler trianglereference triangleorthocenterX_4
Euler trianglereflection triangleorthocenterX_4
Euler trianglesecond Yff circles trianglecenter of first Johnson-Yff circleX_(1478)
excentral triangleextangents triangleBevan pointX_(40)
excentral triangleextouch triangleBevan pointX_(40)
excentral triangleFeuerbach trianglenine-point centerX_5
excentral trianglefirst Brocard triangle**
excentral trianglefirst Morley adjunct trianglesecond Morley-Gibert pointX_(1508)
excentral trianglefirst Morley trianglefirst Morley-Gibert pointX_(1507)
excentral trianglefirst Neuberg triangle**
excentral trianglefirst Yff circles triangleincenterX_1
excentral trianglefirst Yff triangle**
excentral triangleFuhrmann triangleX_(10)-Ceva conjugate of X_1X_(191)
excentral trianglehalf-altitude triangleisogonal conjugate of X_9X_(57)
excentral trianglehexyl trianglecircumcenterX_3
excentral triangleincentral triangleincenterX_1
excentral triangleinner inscribed squares triangle**
excentral triangleinner Napoleon triangle**
excentral triangleinner Vecten triangle**
excentral triangleLemoine triangle**
excentral triangleMacBeath triangleMimosa transform of X_(75)X_(1726)
excentral trianglemedial trianglemittenpunktX_9
excentral trianglemid-arc triangleincenterX_1
excentral trianglemixtilinear triangleincenterX_1
excentral triangleorthic triangleX_4-Ceva conjugate of X_1X_(46)
excentral triangleouter inscribed squares triangle**
excentral triangleouter Napoleon triangle**
excentral triangleouter Vecten triangle**
excentral trianglereference triangleincenterX_1
excentral trianglereflection trianglefirst Evans perspectorX_(484)
excentral trianglesecond Morley adjunct triangle**
excentral trianglesecond Morley triangle**
excentral trianglesecond Neuberg triangle**
excentral trianglesecond Yff circles triangleincenterX_1
excentral trianglesecond Yff triangle**
excentral triangleSteiner triangleX_(99)-Ceva conjugate of X_1X_(1019)
excentral trianglesymmedial triangleX_6-Ceva conjugate of X_1X_(43)
excentral triangletangential mid-arc triangle**
excentral trianglethird Brocard triangle**
excentral trianglethird Morley adjunct triangle**
excentral trianglethird Morley triangle**
excentral triangleYff central trianglecongruent isoscelizers pointX_(173)
excentral triangleYff contact trianglecrossdifference of X_1 and X_2X_(649)
extangents triangleextouch triangleBevan pointX_(40)
extangents triangleFeuerbach triangle**
extangents triangleincentral triangleinternal similitude center of circumcircle and incircleX_(55)
extangents triangleintangents triangleinternal similitude center of circumcircle and incircleX_(55)
extangents triangleorthic triangleClawson pointX_(19)
extangents trianglereference triangleorthocenter of the contact triangleX_(65)
extangents triangletangential triangleinternal similitude center of circumcircle and incircleX_(55)
extouch trianglehalf-altitude triangleincenterX_1
extouch triangleintangents triangleX_8-Ceva conjugate of X_9X_(200)
extouch trianglereference triangleNagel pointX_8
extouch triangletangential triangleX_8-Ceva conjugate of X_6X_(197)
Feuerbach triangleincentral triangleFeuerbach pointX_(11)
Feuerbach trianglereference triangle(X_1,X_5)-harmonic conjugate of X_(11)X_(12)
first Brocard trianglefirst Neuberg trianglecircumcenterX_3
first Brocard trianglesecond Brocard triangletriangle centroidX_2
first Brocard trianglesecond Neuberg trianglecircumcenterX_3
first Brocard trianglethird Brocard triangleEuler line intercept of line X_(32)X_(76)X_(384)
first Morley adjunct trianglefirst Morley trianglefirst Morley centerX_(356)
first Morley trianglesecond Morley trianglesecond Morley-Taylor-Marr centerX_(358)
first Morley trianglethird Morley triangle4th Morley-Taylor-Marr centerX_(1135)
first Neuberg trianglesecond Neuberg trianglecircumcenterX_3
first Yff circles trianglesecond Yff circles triangleincenterX_1
Fuhrmann trianglefirst Brocard trianglecircumcenterX_3
Fuhrmann trianglefirst Neuberg trianglecircumcenterX_3
Fuhrmann triangleinner Napoleon trianglecircumcenterX_3
Fuhrmann triangleinner Vecten trianglecircumcenterX_3
Fuhrmann triangleJohnson trianglecircumcenterX_3
Fuhrmann trianglemedial trianglecircumcenterX_3
Fuhrmann triangleouter Napoleon trianglecircumcenterX_3
Fuhrmann triangleouter Vecten trianglecircumcenterX_3
Fuhrmann trianglesecond Neuberg trianglecircumcenterX_3
Fuhrmann triangletangential trianglecircumcenterX_3
half-altitude triangleinner inscribed squares trianglesymmedian pointX_6
half-altitude trianglemedial trianglesymmedian pointX_6
half-altitude triangleorthic triangleorthocenterX_4
half-altitude triangleouter inscribed squares trianglesymmedian pointX_6
half-altitude trianglereference triangleorthocenterX_4
half-altitude trianglereflection triangleorthocenterX_4
hexyl trianglefirst Yff circles triangle**
hexyl triangleintangents triangleincenterX_1
hexyl trianglereference triangleisogonal conjugate of X_(40)X_(84)
hexyl trianglesecond Yff circles triangle**
hexyl triangletangential mid-arc triangle**
hexyl triangleYff central triangle**
incentral trianglefirst Yff circles triangleincenterX_1
incentral triangleintangents triangleinternal similitude center of circumcircle and incircleX_(55)
incentral trianglemid-arc triangleincenterX_1
incentral trianglemixtilinear triangleincenterX_1
incentral trianglereference triangleincenterX_1
incentral trianglesecond Yff circles triangleincenterX_1
incentral triangletangential triangleinternal similitude center of circumcircle and incircleX_(55)
inner inscribed squares triangleouter inscribed squares trianglesymmedian pointX_6
inner Napoleon trianglefirst Brocard trianglecircumcenterX_3
inner Napoleon trianglefirst Neuberg trianglecircumcenterX_3
inner Napoleon triangleinner Vecten trianglecircumcenterX_3
inner Napoleon triangleJohnson trianglecircumcenterX_3
inner Napoleon trianglemedial trianglecircumcenterX_3
inner Napoleon triangleouter Napoleon trianglecircumcenterX_3
inner Napoleon triangleouter Vecten trianglecircumcenterX_3
inner Napoleon trianglereference trianglesecond Napoleon pointX_(18)
inner Napoleon trianglesecond Neuberg trianglecircumcenterX_3
inner Napoleon triangletangential trianglecircumcenterX_3
inner Vecten trianglefirst Brocard trianglecircumcenterX_3
inner Vecten trianglefirst Neuberg trianglecircumcenterX_3
inner Vecten triangleinner inscribed squares triangleKenmotu pointX_(371)
inner Vecten triangleJohnson trianglecircumcenterX_3
inner Vecten trianglemedial trianglecircumcenterX_3
inner Vecten triangleorthic triangleKenmotu pointX_(371)
inner Vecten triangleouter inscribed squares triangleinner Vecten pointX_(486)
inner Vecten triangleouter Napoleon trianglecircumcenterX_3
inner Vecten triangleouter Vecten trianglecircumcenterX_3
inner Vecten trianglereference triangleinner Vecten pointX_(486)
inner Vecten trianglesecond Neuberg trianglecircumcenterX_3
inner Vecten triangletangential trianglecircumcenterX_3
intangents triangleorthic triangleperspector of the orthic and intangents trianglesX_(33)
intangents triangletangential triangleinternal similitude center of circumcircle and incircleX_(55)
Johnson trianglefirst Brocard trianglecircumcenterX_3
Johnson trianglefirst Neuberg trianglecircumcenterX_3
Johnson trianglefirst Yff circles triangle(X_1,X_5)-harmonic conjugate of X_(11)X_(12)
Johnson triangleMacBeath triangleorthocenterX_4
Johnson trianglemedial trianglecircumcenterX_3
Johnson triangleorthic triangleeigencenter of orthic triangleX_(155)
Johnson triangleouter Napoleon trianglecircumcenterX_3
Johnson triangleouter Vecten trianglecircumcenterX_3
Johnson trianglereference trianglenine-point centerX_5
Johnson trianglesecond Neuberg trianglecircumcenterX_3
Johnson trianglesecond Yff circles triangleFeuerbach pointX_(11)
Johnson triangletangential trianglecircumcenterX_3
Johnson triangleYiu trianglenine-point centerX_5
Lemoine trianglereference triangleisogonal conjugate of X_(574)X_(598)
Lemoine triangletangential triangleorthocorrespondent of X_6X_(1995)
Lucas central trianglefirst Neuberg triangle**
Lucas central triangleLucas inner triangle**
Lucas central triangleLucas tangents triangle**
Lucas central trianglereference trianglecircumcenterX_3
Lucas central trianglesymmedial triangleisogonal conjugate of X_(1131)X_(1151)
Lucas central triangletangential triangleKenmotu pointX_(371)
Lucas inner triangleLucas tangents triangle**
Lucas inner trianglereference triangleisogonal conjugate of X_(1131)X_(1151)
Lucas inner trianglesymmedial triangle**
Lucas inner triangletangential triangle**
Lucas tangents trianglereference triangleKenmotu pointX_(371)
Lucas tangents trianglesymmedial trianglecircumcenterX_3
Lucas tangents triangletangential triangleisogonal conjugate of X_(1131)X_(1151)
MacBeath trianglereference triangleisotomic conjugate of circumcenterX_(264)
MacBeath triangletangential triangleX_6-of-tangential triangleX_(157)
medial trianglefirst Brocard trianglecircumcenterX_3
medial trianglefirst Neuberg trianglecircumcenterX_3
medial trianglefirst Yff circles triangleYff concurrent congruent circles pointX_(498)
medial triangleinner inscribed squares trianglesymmedian pointX_6
medial trianglemid-arc trianglesecond mid-arc pointX_(178)
medial triangleouter inscribed squares trianglesymmedian pointX_6
medial triangleouter Napoleon trianglecircumcenterX_3
medial triangleouter Vecten trianglecircumcenterX_3
medial trianglereference triangletriangle centroidX_2
medial trianglesecond Neuberg trianglecircumcenterX_3
medial trianglesecond Yff circles triangle(X_1,X_2)-harmonic conjugate of X_(498)X_(499)
medial triangletangential trianglecircumcenterX_3
mid-arc trianglefirst Yff circles triangleincenterX_1
mid-arc trianglemixtilinear triangleincenterX_1
mid-arc trianglereference triangleincenterX_1
mid-arc trianglesecond Yff circles triangleincenterX_1
mid-arc triangletangential mid-arc triangle**
mixtilinear trianglefirst Yff circles triangleincenterX_1
mixtilinear trianglereference triangleincenterX_1
mixtilinear trianglesecond Yff circles triangleincenterX_1
orthic triangleinner inscribed squares triangleKenmotu pointX_(371)
orthic triangleouter inscribed squares triangle(X_3,X_6)-harmonic conjugate of X_(371)X_(372)
orthic triangleouter Vecten triangle(X_3,X_6)-harmonic conjugate of X_(371)X_(372)
orthic trianglereference triangleorthocenterX_4
orthic trianglereflection triangleorthocenterX_4
orthic triangletangential trianglehomothetic center of orthic and tangential trianglesX_(25)
outer Napoleon trianglefirst Brocard trianglecircumcenterX_3
outer Napoleon trianglefirst Neuberg trianglecircumcenterX_3
outer Napoleon triangleouter Vecten trianglecircumcenterX_3
outer Napoleon trianglereference trianglefirst Napoleon pointX_(17)
outer Napoleon trianglesecond Neuberg trianglecircumcenterX_3
outer Napoleon triangletangential trianglecircumcenterX_3
outer Vecten trianglefirst Brocard trianglecircumcenterX_3
outer Vecten trianglefirst Neuberg trianglecircumcenterX_3
outer Vecten triangleinner inscribed squares triangleouter Vecten pointX_(485)
outer Vecten triangleouter inscribed squares triangle(X_3,X_6)-harmonic conjugate of X_(371)X_(372)
outer Vecten trianglereference triangleouter Vecten pointX_(485)
outer Vecten trianglesecond Neuberg trianglecircumcenterX_3
outer Vecten triangletangential trianglecircumcenterX_3
reference trianglefirst Brocard trianglethird Brocard pointX_(76)
reference trianglefirst Morley adjunct trianglesecond Morley-Taylor-Marr centerX_(358)
reference trianglefirst Morley trianglefirst Morley-Taylor-Marr centerX_(357)
reference trianglefirst Neuberg triangleTarry pointX_(98)
reference trianglefirst Yff circles triangleincenterX_1
reference trianglefirst Yff triangle**
reference triangleinner inscribed squares triangleouter Vecten pointX_(485)
reference triangleouter inscribed squares triangleinner Vecten pointX_(486)
reference trianglereflection triangleorthocenterX_4
reference trianglesecond Brocard trianglesymmedian pointX_6
reference trianglesecond Morley adjunct triangle6th Morley-Taylor-Marr centerX_(1137)
reference trianglesecond Morley triangle5th Morley-Taylor-Marr centerX_(1136)
reference trianglesecond Neuberg triangleisogonal conjugate of X_(182)X_(262)
reference trianglesecond Yff circles triangleincenterX_1
reference trianglesecond Yff triangle**
reference triangleSteiner triangleSteiner pointX_(99)
reference trianglesymmedial trianglesymmedian pointX_6
reference triangletangential mid-arc trianglefirst mid-arc pointX_(177)
reference triangletangential trianglesymmedian pointX_6
reference trianglethird Brocard trianglethird power pointX_(32)
reference trianglethird Morley adjunct triangle4th Morley-Taylor-Marr centerX_(1135)
reference trianglethird Morley trianglethird Morley-Taylor-Marr centerX_(1134)
reference triangleYff central trianglefirst mid-arc pointX_(177)
reference triangleYff contact triangleYff parabolic pointX_(190)
reference triangleYiu trianglenine-point centerX_5
second Morley adjunct trianglesecond Morley triangle**
second Morley trianglethird Morley triangle6th Morley-Taylor-Marr centerX_(1137)
Stammler trianglefirst Morley triangle**
Stammler trianglesecond Morley triangle**
Stammler trianglethird Morley triangle**
Steiner triangletangential trianglecrossdifference of X_2 and X_(39)X_(669)
symmedial trianglesecond Brocard trianglesymmedian pointX_6
symmedial triangletangential trianglesymmedian pointX_6
symmedial trianglethird Brocard triangleX_6-Ceva conjugate of X_2X_(194)
tangential mid-arc triangleYff central trianglefirst mid-arc pointX_(177)
tangential trianglefirst Brocard trianglecircumcenterX_3
tangential trianglefirst Neuberg trianglecircumcenterX_3
tangential trianglefirst Yff triangle**
tangential trianglesecond Brocard trianglesymmedian pointX_6
tangential trianglesecond Neuberg trianglecircumcenterX_3
tangential trianglesecond Yff triangle**
tangential trianglethird Brocard triangle**
tangential triangleYff contact triangle**
tangential triangleYiu triangleX_5-Ceva conjugate of X_3X_(195)
third Brocard trianglefirst Neuberg triangleorthocenterX_4
third Morley adjunct trianglethird Morley triangle**

See also

Perspective Center, Perspective Triangles, Perspectrix

Portions of this entry contributed by Floor van Lamoen

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References

Kimberling, C. "Encyclopedia of Triangle Centers Glossary." http://faculty.evansville.edu/ck6/encyclopedia/glossary.html.

Referenced on Wolfram|Alpha

Perspector

Cite this as:

van Lamoen, Floor and Weisstein, Eric W. "Perspector." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Perspector.html

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