Eigencenter

The eigencenter of a central triangle is the perspector of and its unary cofactor triangle , which are perspective.

Let the -, -, and -vertices of have trilinear coordinates for , 2, 3. Also define

			(1)
			(2)
			(3)
			(4)

Also define

(5)

and and cyclically. Then the eigencenter of is the point .

The following table summarizes eigencenters of named triangles that are Kimberling centers.

triangle	Kimberling center	eigencenter
anticomplementary triangle		-Ceva conjugate of
circumcircle mid-arc triangle		psi(incenter, symmedian point)
circum-medial triangle		Steiner point
circumnormal triangle		psi(orthocenter, symmedian point)
circumtangential triangle		Parry point
contact triangle		-Ceva conjugate of
excentral triangle		incenter
extouch triangle		eigentransform of
incentral triangle		incenter
inner Napoleon triangle		first isodynamic point
medial triangle		circumcenter
orthic triangle		eigencenter of orthic triangle
outer Napoleon triangle		second isodynamic point
second Brocard triangle		triangle centroid
Stammler triangle		symmedian point
Steiner triangle		Brocard midpoint
symmedial triangle		-Ceva conjugate of
tangential triangle		circumcenter
Yff contact triangle		incenter