Given a reference triangle and a point , the triple , with , and representing the distances from to the vertices of the reference triangle, is the tripolar coordinates of .
The tripolar coordinates satisfy
(Euler 1786).
Given , the number of points having tripolar coordinates satisfying depends on , and being the sides of a triangle (two points), a degenerate triangle (one point) or not a triangle (zero points) (Bottema 1987)
The following table summarizes the tripolar coordinated for a number of named centers.
center | tripolar coordinates | |
incenter | ||
triangle centroid | ||
circumcenter | ||
orthocenter | ||
symmedian point |