Proportionally cutting circles are circles that intersect the sidelines of a reference triangle such that length of the chords that are cut off have lengths , , and that are proportional to the corresponding side lengths , , and of .
The circumcircle and Stammler circles are proportionally-cutting circles. Let be the vertex of the anticevian triangle of the circumcenter , then the circle with center passing through is a proportionally cutting circle, and similarly for and .
The centers of proportionally-cutting circles lie on the Stammler hyperbola.
Depending on , there may be 2, 3 or 4 proportionally-cutting circles (Stammler 1992). If is a value giving four proportionally-cutting circles, then the centers of these circles form an orthocentric system of which the circumcircle is the nine-point circle (Ehrmann and van Lamoen 2002).