Proportionally cutting circles are circles that intersect the sidelines of a reference triangle Δ A B C such that length of the chords that are cut off have lengths μ a, μ b, and μ c that are proportional to the corresponding side lengths a , b, and c of Δ A B C. The circumcircle and Stammler circles are proportionally-cutting circles. Let O_A be the A - vertex of the anticevian triangle of the circumcenter O, then the circle with center O_A passing through A is a proportionally cutting circle, and similarly for O_B and O_C. The centers of proportionally-cutting circles lie on the Stammler hyperbola.
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