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 |