Cevian Triangle

Given a point and a triangle , the Cevian triangle is defined as the triangle composed of the endpoints of the cevians though the Cevian point . A triangle and its Cevian triangle are therefore perspective with respect to the Cevian point. If the point has trilinear coordinates , then the Cevian triangle has trilinear vertex matrix

(1)

(Kimberling 1998, pp. 55 and 185), and is a central triangle of type 1 (Kimberling 1998, p. 55).

The following table summarizes a number of special Cevian triangles for various special Cevian points .

Cevian point	Kimberling center	Cevian triangle
incenter		incentral triangle
triangle centroid		medial triangle
orthocenter		orthic triangle
symmedian point		symmedial triangle
Gergonne point		contact triangle
Nagel point		extouch triangle
Steiner point		Steiner triangle
Yff parabolic point		Yff contact triangle
		MacBeath triangle
		Lemoine triangle

If is the Cevian triangle of and is the anticevian triangle, then and are harmonic conjugates with respect to and .

The side lengths of the Cevian triangle with respect to a Cevian point are given by

			(2)
			(3)
			(4)

The area of the Cevian triangle of with respect to the center with trilinear coordinates is given by

(5)

where is the area of triangle .

If is the Cevian triangle of , then the triangle obtained by reflecting , , and across the midpoints of their sides is also a Cevian triangle of (Honsberger 1995, p. 141; left figure). Furthermore, if the Cevian circle crosses the sides of in three points , , and , then is also a Cevian triangle of (Honsberger 1995, pp. 141-142; right figure).