1985 IMO Problems/Problem 5
A circle with center passes through the vertices and of the triangle and intersects the segments and again at distinct points and respectively. Let be the point of intersection of the circumcircles of triangles and (apart from ). Prove that .
is the Miquel Point of quadrilateral , so there is a spiral similarity centered at that takes to . Let be the midpoint of and be the midpoint of . Thus the spiral similarity must also send to and must also be the center of another spiral similarity that sends to , so is cyclic. is also cyclic with diameter and thus must lie on the same circumcircle as , , and so .