Extouch Triangle

The extouch triangle is the triangle formed by the points of tangency of a triangle with its excircles , , and . The points , , and can also be constructed as the points which bisect the perimeter of starting at , , and .

It is the Cevian triangle of the Nagel point (Kimberling 1998, p. 158), the pedal triangle of the Bevan point , and the cyclocevian triangle of .

It is the polar triangle of the Mandart inellipse.

Equivalent trilinear vertex matrices for the extouch triangle are

			(1)
			(2)

The lengths of the sides of the extouch triangle are

			(3)
			(4)
			(5)
			(6)
			(7)
			(8)

where is the triangle area.

The extouch triangle has triangle area

			(9)
			(10)

where , , and are the triangle area, inradius, and semiperimeter, respectively, of the original triangle . This is the same area as the contact triangle.

The circumcircle of the extouch triangle is known as the Mandart circle.

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

	center of extouch triangle		center of reference triangle
	triangle centroid		-Ceva conjugate of
	circumcenter		circumcenter of extouch triangle