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Perspectrix


PerspectiveTriangles

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 1triangle 2perspectrixKimberlingline
anticomplementary trianglecircum-medial trianglethird power pointL_(32)de Longchamps line
anticomplementary trianglecircum-orthic triangle***
anticomplementary trianglecontact triangle***
anticomplementary triangleD-triangle***
anticomplementary triangleextouch triangle***
anticomplementary trianglefirst Brocard triangle***
anticomplementary trianglefirst Neuberg triangle***
anticomplementary trianglefirst Yff triangle***
anticomplementary triangleFuhrmann triangle***
anticomplementary triangleincentral triangleX_6-Ceva conjugate of X_1L_(43)
anticomplementary triangleinner Napoleon triangle***
anticomplementary triangleinner Vecten triangle***
anticomplementary triangleLemoine triangle***
anticomplementary triangleMacBeath trianglethird power pointL_(32)de Longchamps line
anticomplementary triangleorthic triangle***
anticomplementary triangleouter Napoleon triangle***
anticomplementary triangleouter Vecten triangle***
anticomplementary trianglesecond Neuberg triangle***
anticomplementary trianglesecond Yff triangle***
anticomplementary triangleSteiner triangle***
anticomplementary trianglesymmedial triangleX_6-Ceva conjugate of X_2L_(194)
anticomplementary trianglethird Brocard triangle***
anticomplementary triangleYff contact triangleexsimilicenter of Moses circle and incircleL_(1015)
BCI triangleexcentral triangle***
BCI trianglereference triangle***
BCI triangletangential mid-arc triangle***
circumcircle mid-arc trianglefirst Brocard triangle***
circumcircle mid-arc trianglefirst Neuberg triangle***
circumcircle mid-arc trianglefirst Yff circles triangle***
circumcircle mid-arc triangleFuhrmann trianglepsi(incenter, symmedian point)L_(101)
circumcircle mid-arc triangleincentral trianglemittenpunktL_9
circumcircle mid-arc triangleinner Napoleon triangle***
circumcircle mid-arc triangleinner Vecten triangle***
circumcircle mid-arc triangleJohnson triangle***
circumcircle mid-arc trianglemedial triangleX_9-Ceva conjugate of X_6L_(198)
circumcircle mid-arc trianglemid-arc triangle***
circumcircle mid-arc trianglemixtilinear trianglecrosspoint of X_2 and X_(84)L_(1108)
circumcircle mid-arc triangleouter Napoleon triangle***
circumcircle mid-arc triangleouter Vecten triangle***
circumcircle mid-arc trianglereference trianglecrosspoint of incenter and triangle centroidL_(37)
circumcircle mid-arc trianglesecond Neuberg triangle***
circumcircle mid-arc trianglesecond Yff circles triangle***
circumcircle mid-arc triangletangential trianglemittenpunktL_9
circum-medial triangleD-triangle***
circum-medial triangleMacBeath trianglethird power pointL_(32)de Longchamps line
circum-medial trianglemedial trianglecircumcenterL_3orthic axis
circum-medial trianglereference triangleBrocard midpointL_(39)
circum-medial triangletangential trianglecircumcenterL_3orthic axis
circum-orthic triangleEuler trianglecircumcenterL_3orthic axis
circum-orthic trianglehalf-altitude triangle***
circum-orthic triangleJohnson triangle***
circum-orthic triangleMacBeath triangleharmonic of X_(216)L_(577)
circum-orthic trianglereference triangleX_5-Ceva conjugate of X_(51)L_(216)
circum-orthic trianglereflection triangle***
contact trianglefirst Yff circles triangle***
contact trianglehalf-altitude triangle***
contact triangleintangents triangleX_7-Ceva conjugate of X_(55)L_(218)
contact trianglemid-arc triangleisogonal conjugate of X_(174)L_(259)
contact trianglereference triangleinternal similitude center of circumcircle and incircleL_(55)Gergonne line
contact trianglesecond Yff circles triangle***
contact triangletangential triangle***
D-triangleinner Napoleon trianglefocus of Kiepert parabolaL_(110)
D-trianglemedial triangletangential isogonal conjugate of X_(23)L_(2930)
D-triangleouter Napoleon trianglefocus of Kiepert parabolaL_(110)
D-trianglereference triangleisogonal conjugate of X_(468)L_(895)
Euler trianglehalf-altitude triangle***
Euler triangleorthic triangleharmonic of X_(216)L_(577)
Euler trianglereflection triangle***
excentral triangleextangents triangleincenterL_1antiorthic axis
excentral triangleextouch triangleX_1-Ceva conjugate of X_(56)L_(221)
excentral triangleFeuerbach triangleincenterL_1antiorthic axis
excentral trianglefirst Brocard triangle***
excentral trianglefirst Morley adjunct triangle***
excentral trianglefirst Morley triangle***
excentral trianglefirst Neuberg triangle***
excentral trianglefirst Yff circles triangle***
excentral trianglefirst Yff triangle***
excentral triangleFuhrmann triangle***
excentral trianglehalf-altitude triangleX_1-Ceva conjugate of X_(56)L_(221)
excentral triangleincentral triangleincenterL_1antiorthic axis
excentral triangleinner inscribed squares triangle***
excentral triangleinner Napoleon triangle***
excentral triangleinner Vecten triangle***
excentral triangleLemoine triangle***
excentral triangleMacBeath triangle***
excentral trianglemedial triangleinternal similitude center of circumcircle and incircleL_(55)Gergonne line
excentral trianglemid-arc triangle***
excentral trianglemixtilinear triangle***
excentral triangleorthic triangle***
excentral triangleouter inscribed squares trianglepoint Caroli IIL_(2067)
excentral triangleouter Napoleon triangle***
excentral triangleouter Vecten triangle***
excentral trianglereference triangleincenterL_1antiorthic axis
excentral trianglereflection triangle***
excentral trianglesecond Morley adjunct triangle***
excentral trianglesecond Morley triangle***
excentral trianglesecond Neuberg triangle***
excentral trianglesecond Yff circles triangle***
excentral trianglesecond Yff triangle***
excentral triangleSteiner triangleexsimilicenter of Moses circle and incircleL_(1015)
excentral trianglesymmedial triangleX_1-Ceva conjugate of X_2L_(192)
excentral trianglethird Brocard triangle***
excentral trianglethird Morley adjunct triangle***
excentral trianglethird Morley triangle***
excentral triangleYff contact triangle***
extangents triangleextouch triangleX_7-Ceva conjugate of X_(56)L_(222)
extangents triangleFeuerbach triangleincenterL_1antiorthic axis
extangents triangleincentral triangleincenterL_1antiorthic axis
extangents trianglereference triangleincenterL_1antiorthic axis
extouch trianglehalf-altitude triangleX_1-Ceva conjugate of X_(56)L_(221)
extouch triangleintangents triangle***
extouch trianglereference triangleexternal similitude center of circumcircle and incircleL_(56)
extouch triangletangential trianglecenter of Yiu conicL_(478)
Feuerbach triangleincentral triangleincenterL_1antiorthic axis
Feuerbach trianglereference triangleincenterL_1antiorthic axis
first Brocard trianglefirst Neuberg triangleisogonal conjugate of X_(385)L_(694)
first Brocard trianglesecond Brocard triangle***
first Brocard trianglesecond Neuberg triangle***
first Brocard trianglethird Brocard triangleisogonal conjugate of X_(385)L_(694)
first Morley adjunct trianglefirst Morley triangle***
first Neuberg trianglesecond Neuberg triangle***
Fuhrmann trianglefirst Brocard triangle***
Fuhrmann trianglefirst Neuberg triangle***
Fuhrmann triangleinner Napoleon triangle***
Fuhrmann triangleinner Vecten triangle***
Fuhrmann triangleJohnson triangle***
Fuhrmann trianglemedial triangle***
Fuhrmann triangleouter Napoleon triangle***
Fuhrmann triangleouter Vecten triangle***
Fuhrmann trianglesecond Neuberg triangle***
Fuhrmann triangletangential triangle***
half-altitude triangleinner inscribed squares triangleisogonal conjugate of X_(1132)L_(1152)
half-altitude trianglemedial triangleX_3-Ceva conjugate of X_6L_(154)
half-altitude triangleorthic trianglereflection of X_(64) in X_3L_(1498)
half-altitude triangleouter inscribed squares triangleisogonal conjugate of X_(1131)L_(1151)
half-altitude trianglereference triangleisogonal conjugate of X_(20)L_(64)
half-altitude trianglereflection triangle***
hexyl trianglefirst Yff circles triangle***
hexyl triangleintangents trianglemittenpunktL_9
hexyl trianglereference triangleClawson pointL_(19)
hexyl trianglesecond Yff circles triangle***
incentral trianglefirst Yff circles triangle***
incentral triangleintangents triangleMimosa transform of X_(222)L_(1743)
incentral trianglemid-arc triangle***
incentral trianglemixtilinear triangle***
incentral trianglereference triangleincenterL_1antiorthic axis
incentral trianglesecond Yff circles triangle***
incentral triangletangential trianglemittenpunktL_9
inner inscribed squares triangleouter inscribed squares trianglecircumcenterL_3orthic axis
inner Napoleon trianglefirst Brocard triangle***
inner Napoleon trianglefirst Neuberg triangle***
inner Napoleon triangleinner Vecten triangle***
inner Napoleon triangleJohnson triangle***
inner Napoleon trianglemedial triangle***
inner Napoleon triangleouter Napoleon trianglefocus of Kiepert parabolaL_(110)
inner Napoleon triangleouter Vecten triangle***
inner Napoleon trianglereference triangle***
inner Napoleon trianglesecond Neuberg triangle***
inner Napoleon triangletangential triangle***
inner Vecten trianglefirst Brocard triangle***
inner Vecten trianglefirst Neuberg triangle***
inner Vecten triangleinner inscribed squares triangle***
inner Vecten triangleJohnson triangle***
inner Vecten trianglemedial triangle***
inner Vecten triangleorthic triangle***
inner Vecten triangleouter inscribed squares triangle***
inner Vecten triangleouter Napoleon triangle***
inner Vecten triangleouter Vecten triangle***
inner Vecten trianglereference trianglesecond van Lamoen homothetic centerL_(494)
inner Vecten trianglesecond Neuberg triangle***
inner Vecten triangletangential triangle***
Johnson trianglefirst Brocard triangle***
Johnson trianglefirst Neuberg triangle***
Johnson triangleMacBeath triangleorthoharmonic of X_(52)L_(571)
Johnson triangleorthic triangle***
Johnson triangleouter Napoleon triangle***
Johnson triangleouter Vecten triangle***
Johnson trianglesecond Neuberg triangle***
Johnson triangletangential triangleharmonic of X_(216)L_(577)
Johnson triangleYiu triangle***
Lemoine trianglereference triangleharmonic of X_(187)L_(574)
Lemoine triangletangential triangle***
Lucas central trianglefirst Neuberg triangle***
Lucas central triangleLucas inner triangletriangle centroidL_2Lemoine axis
Lucas central triangleLucas tangents triangletriangle centroidL_2Lemoine axis
Lucas central trianglereference triangletriangle centroidL_2Lemoine axis
Lucas central trianglesymmedial triangletriangle centroidL_2Lemoine axis
Lucas central triangletangential triangletriangle centroidL_2Lemoine axis
Lucas inner triangleLucas tangents triangletriangle centroidL_2Lemoine axis
Lucas inner trianglereference triangletriangle centroidL_2Lemoine axis
Lucas inner trianglesymmedial triangletriangle centroidL_2Lemoine axis
Lucas inner triangletangential triangletriangle centroidL_2Lemoine axis
Lucas tangents trianglereference triangletriangle centroidL_2Lemoine axis
Lucas tangents trianglesymmedial triangletriangle centroidL_2Lemoine axis
Lucas tangents triangletangential triangletriangle centroidL_2Lemoine axis
MacBeath trianglereference triangleinverse of X_(125) in the Brocard circleL_(184)
MacBeath triangletangential triangle***
medial trianglefirst Brocard triangle***
medial trianglefirst Neuberg triangleTCC-perspector of X_(31)L_(1613)
medial triangleinner inscribed squares triangle***
medial trianglemid-arc triangle***
medial triangleouter inscribed squares triangle***
medial triangleouter Napoleon triangle***
medial triangleouter Vecten triangle***
medial trianglesecond Neuberg triangle***
medial triangletangential trianglecircumcenterL_3orthic axis
mid-arc trianglefirst Yff circles triangle***
mid-arc trianglemixtilinear triangle***
mid-arc trianglereference triangle***
mid-arc trianglesecond Yff circles triangle***
mid-arc triangletangential mid-arc triangle***
mixtilinear trianglefirst Yff circles triangle***
mixtilinear trianglereference trianglemittenpunktL_9
mixtilinear trianglesecond Yff circles triangle***
orthic triangleinner inscribed squares trianglecircumcenterL_3orthic axis
orthic triangleouter inscribed squares trianglecircumcenterL_3orthic axis
orthic triangleouter Vecten triangle***
orthic trianglereference trianglecircumcenterL_3orthic axis
orthic trianglereflection triangleX_5-Ceva conjugate of X_3L_(195)
outer Napoleon trianglefirst Brocard triangle***
outer Napoleon trianglefirst Neuberg triangle***
outer Napoleon triangleouter Vecten triangle***
outer Napoleon trianglereference triangleDC(X_(13))L_(2981)
outer Napoleon trianglesecond Neuberg triangle***
outer Napoleon triangletangential triangle***
outer Vecten trianglefirst Brocard triangle***
outer Vecten trianglefirst Neuberg triangle***
outer Vecten triangleinner inscribed squares triangle***
outer Vecten triangleouter inscribed squares triangle***
outer Vecten trianglereference trianglefirst van Lamoen homothetic centerL_(493)
outer Vecten trianglesecond Neuberg triangle***
outer Vecten triangletangential triangle***
reference trianglefirst Brocard triangleisogonal conjugate of X_(385)L_(694)
reference trianglefirst Morley adjunct triangle***
reference trianglefirst Morley triangle***
reference trianglefirst Neuberg triangleisogonal conjugate of X_(385)L_(694)
reference trianglefirst Yff triangle***
reference triangleinner inscribed squares trianglecircumcenterL_3orthic axis
reference triangleouter inscribed squares trianglecircumcenterL_3orthic axis
reference trianglereflection triangleKosnita pointL_(54)
reference trianglesecond Brocard triangleParry pointL_(111)
reference trianglesecond Morley adjunct triangle***
reference trianglesecond Morley triangle***
reference trianglesecond Neuberg triangle***
reference trianglesecond Yff triangle***
reference triangleSteiner triangleisogonal conjugate of X_(99)L_(512)
reference trianglesymmedial triangletriangle centroidL_2Lemoine axis
reference triangletangential mid-arc triangleisogonal conjugate of X_(174)L_(259)
reference triangletangential triangletriangle centroidL_2Lemoine axis
reference trianglethird Brocard triangleisogonal conjugate of X_(385)L_(694)
reference trianglethird Morley adjunct triangle***
reference trianglethird Morley triangle***
reference triangleYff central triangle***
reference triangleYff contact trianglecrossdifference of X_1 and X_2L_(649)Nagel line
reference triangleYiu trianglebarycentric product X_(17)*X_(18)L_(2963)
second Morley adjunct trianglesecond Morley triangle***
Steiner triangletangential trianglecrosspoint of X_2 and X_(512)L_(1084)
symmedial trianglesecond Brocard triangle***
symmedial triangletangential triangletriangle centroidL_2Lemoine axis
symmedial trianglethird Brocard triangle***
tangential trianglefirst Brocard triangle***
tangential trianglefirst Neuberg triangle***
tangential trianglefirst Yff triangle***
tangential trianglesecond Brocard triangle***
tangential trianglesecond Neuberg triangle***
tangential trianglesecond Yff triangle***
tangential trianglethird Brocard triangle***
tangential triangleYff contact triangle***
tangential triangleYiu triangle***
third Brocard trianglefirst Neuberg triangleisogonal conjugate of X_(385)L_(694)
third Morley adjunct trianglethird Morley triangle***

See also

Perspector, Perspective Triangles, Sondat's Theorem

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References

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

Referenced on Wolfram|Alpha

Perspectrix

Cite this as:

Weisstein, Eric W. "Perspectrix." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Perspectrix.html

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