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).
