GeezerVison is showing mathworld.wolfram.com — Contact Triangle -- from Wolfram MathWorld

Contact Triangle

The contact triangle of a triangle , also called the intouch triangle, is the triangle formed by the points of tangency of the incircle of with .

The contact triangle is therefore the pedal triangle of with respect to the incenter of . It is also the Cevian triangle of with respect to the Gergonne point Ge (Kimberling 1998, p. 158) and the cyclocevian triangle with respect to the same point.

The contact triangle is the polar triangle of the incircle.

The contact triangle has equivalent trilinear vertex matrices

			(1)
			(2)

The side lengths of are

			(3)
			(4)
			(5)

The area is given by

			(6)
			(7)
			(8)

where , , , and are the area, inradius, semiperimeter, and circumradius, respectively, of the reference triangle . This is the same area as the extouch triangle.

Beginning with an arbitrary triangle , find the contact triangle . Then find the contact triangle of that triangle, and so on. Then the resulting triangle approaches an equilateral triangle (Goldoni 2003). The analogous result also holds for iterative construction of excentral triangles (Johnson 1929, p. 185; Goldoni 2003).

The Gergonne point Ge of is equivalent to the symmedian point of .

The following table gives selected Kimberling centers with of the contact triangle in terms of the centers of the reference triangle (Kimberling). The correspondences hold wherever both centers are defined. The row for identifies the same point at infinity in the two triangles.

	center of contact triangle		center of reference triangle
	incenter		first mid-arc point
	triangle centroid		Weill point
	circumcenter		incenter
	orthocenter		orthocenter of the intouch triangle
	nine-point center		inverse-in-incircle of
	symmedian point		Gergonne point
	Clawson point		second Stevanovic perspector
	Exeter point		internal similitude center of circumcircle and incircle
	far-out point		Schröder point
	perspector of abc and orthic-of-orthic triangle		external similitude center of circumcircle and incircle
	homothetic center of orthic and tangential triangles		isogonal conjugate of
	circumcenter of the tangential triangle		circumcenter
	Euler infinity point		isogonal conjugate of
	second power point		-Ceva conjugate of
	triangle centroid of orthic triangle		point Maia II
	internal similitude center of circumcircle and incircle		Yff center of congruence
	external similitude center of circumcircle and incircle		third mid-arc point
	isogonal conjugate of		anticomplement of Nagel point
			reflection of in
	Tarry point		Brisse transform of
	Steiner point		6th Stevanovic point

Explore with Wolfram|Alpha

More things to try:

References

Danneels, E. "The Intouch Triangle and the OI-Line." Forum Geometricorum 4, 125-134, 2004. https://web.archive.org/web/20240406003845/https://forumgeom.fau.edu/FG2004volume4/FG200416index.html.Goldoni, G. "Problem 10993." Amer. Math. Monthly 110, 155, 2003.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929.Kimberling, C. "Encyclopedia of Triangle Centers." https://faculty.evansville.edu/ck6/encyclopedia/ETC.html, https://faculty.evansville.edu/ck6/encyclopedia/ETCPart2.html, and https://faculty.evansville.edu/ck6/encyclopedia/ETCPart3.html.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.Oldknow, A. "The Euler-Gergonne-Soddy Triangle of a Triangle." Amer. Math. Monthly 103, 319-329, 1996.

Referenced on Wolfram|Alpha

Contact Triangle

Cite this as:

Weisstein, Eric W. "Contact Triangle." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ContactTriangle.html

GeezerVison

Contact Triangle

See also

Explore with Wolfram|Alpha

References

Referenced on Wolfram|Alpha

Cite this as:

Subject classifications