Cyclocevian Conjugate

Let be a point not on a sideline of a reference triangle . Let be the point of intersection , , and . The circle through meets each sideline in two (not necessarily distinct) points , ; , ; and , . Then the lines , , and concur in a point known as the cyclocevian conjugate of . The point has triangle center function

(Kimberling 1998, p. 226).

The Gergonne point is its own cyclocevian conjugate.

The following table summarized cyclocevian conjugates for some triangle centers.

Kimberling	center	cyclocevian conjugate	name
	incenter
	triangle centroid		orthocenter
	orthocenter		triangle centroid
	symmedian point
	Gergonne point		Gergonne point
	Nagel point
	de Longchamps point
	isogonal conjugate of
	symmedian point of the anticomplementary triangle
			Nagel point
			symmedian point of the anticomplementary triangle
			incenter
			symmedian point
			de Longchamps point
			isotomic conjugate of
			isogonal conjugate of