Tangential Triangle

The tangential triangle is the triangle formed by the lines tangent to the circumcircle of a given triangle at its vertices. It is therefore antipedal triangle of with respect to the circumcenter . It is also anticevian triangle of with the symmedian point as the anticevian point (Kimberling 1998, p. 156). Furthermore, the symmedian point of is the Gergonne point of .

The tangential triangle is the polar triangle of the circumcircle.

Its trilinear vertex matrix is

(1)

The side lengths of the tangential triangle are

			(2)
			(3)
			(4)

For an acute triangle, the perimeter is given by

(5)

and the product of side lengths is

(6)

The area of the tangential triangle is

(7)

where is the triangle area of .

The following table gives the centers of the tangential triangle in terms of the centers of the reference triangle that correspond to Kimberling centers .

	center of tangential triangle		center of reference triangle
	triangle centroid		-Ceva conjugate of
	circumcenter		circumcenter of the tangential triangle
	orthocenter		eigencenter of orthic triangle
	nine-point center		-of-tangential-triangle
	symmedian point		-of-tangential-triangle
	Euler infinity point		isogonal conjugate of
	isogonal conjugate of		Napoleon crossdifference

Given a triangle and its tangential triangle , the extensions of the sides of the two triangles intersect in three points , , and , which are collinear (Honsberger 1995).

The sides of an orthic triangle are parallel to the tangents to the circumcircle at the vertices (Johnson 1929, p. 172). This is equivalent to the statement that each line from a triangle's circumcenter to a vertex is always perpendicular to the corresponding side of the orthic triangle (Honsberger 1995, p. 22), and to the fact that the orthic and tangential triangles are homothetic.