Let the inner and outer Soddy triangles of a reference triangle be denoted and , respectively. Similarly, let the tangential triangles of and be denoted and , respectively. Then the inner (respectively, outer) Rigby point Ri (respectively, ) is the perspector of and (respectively, and ) (Oldknow 1996). The Rigby points lie on the Soddy line. They have triangle center functions
which are Kimberling centers and , respectively.
Honsberger (1995) defines a different point which he calls the "Rigby point" . Let be an arbitrary chord of the circumcircle of a given triangle , and let be the Simson line pole of the Simson line with respect to which is perpendicular to . Then it also turns out that and . In addition, , , and with respect to .
As a result of these remarkable facts, it can be shown that the Simson lines , , and with respect to meet in the Rigby point . Moreover, the Simson lines , , and with respect to also meet in , and is the orthopole of , , and with respect to , and of , , and with respect to . Finally, is the midpoint of the orthocenters of and (Honsberger 1996, p. 136).