The Triangle Formed by the Centers of the Miquel Circles
Let ABC be a triangle. Let A', B', and C' be points on BC, AC, and AB, respectively. Let A'', B'', and C'' be the circumcenters of AB'C', BA'C', and CA'B', respectively. Then A"B''C'' is similar to ABC.
See problem 12 in D. R. Davis, Modern College Geometry, Cambridge, MA: Addison-Wesley Press, 1949 p. 161.