Tangential Triangle

The tangential triangle is the triangle formed by the tangent lines to the circumcircle of a given triangle at its vertices. It is therefore the antipedal triangle of with respect to the circumcenter . It is also the 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 when is an acute triangle.

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 triangle 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 . An asterisk marks a correspondence stated for an acute triangle as the reference triangle. All correspondences are understood wherever the indicated triangles and centers are defined. Additional correspondences were communicated by D. Reznik (pers. comm., Feb. 4, 2021).

	center of tangential triangle		center of reference triangle
	incenter		circumcenter
	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
	Gergonne point		symmedian point
	Nagel point		reflection of in
	mittenpunkt		-of-tangential-triangle
	Spieker center		Spieker center of the tangential triangle
	Feuerbach point		focus of the Kiepert parabola
	(,)-harmonic conjugate of		perspector of the tangential triangle and circumcevian triangle of
	de Longchamps point		de Longchamps point of the tangential triangle
	Schiffler point		isogonal conjugate of reference with respect to the tangential triangle
	Euler infinity point		isogonal conjugate of
	second power point		perspector of the tangential triangle and circumcevian triangle of
	(,)-harmonic conjugate of		isogonal conjugate of reference with respect to the tangential triangle
	inverse point of the incenter in the circumcircle		inverse point of in the circumcircle
	crosspoint of incenter and triangle centroid		-of-tangential-triangle
	Bevan point		intersection of the Euler line and
	-Ceva conjugate of		(,)-harmonic conjugate of
	internal similitude center(circumcircle, incircle)		Exeter point
	external similitude center(circumcircle, incircle)		perspector of and the orthic triangle of its orthic triangle
	isogonal conjugate of		homothetic center of the orthic triangle and tangential triangle
	isogonal conjugate of		perspector of the reflection of in its Euler line and the tangential triangle
	isogonal conjugate of		perspector of the tangential triangle and circumcevian triangle of
	isogonal conjugate of		-of-tangential-triangle
	orthocenter of the contact triangle		orthocenter
	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.