Incentral Triangle

The incentral triangle is the Cevian triangle of a triangle with respect to its incenter . It is therefore also the triangle whose vertices are determined by the intersections of the reference triangle's angle bisectors with the respective opposite sides.

Its trilinear vertex matrix is

(1)

It is perspective to every anticevian triangle (Kimberling 1998, p. 157).

It is the cyclocevian triangle with respect to Kimberling center .

The side lengths of the incentral triangle are

			(2)
			(3)
			(4)

and its area is

(5)

where is the area of the reference triangle.

The circumcircle of the incentral triangle is the incentral circle.

The following table gives the centers of the incentral triangle in terms of the centers of the reference triangle that are Kimberling centers .

	center of incentral triangle		center of reference triangle
	triangle centroid		bicentric sum of pu(32)
	orthocenter		orthocenter of the incentral triangle

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