The Circumcircles of Four Triangles
Let ABC be a triangle. Let DEF be parallel to AC with D on AB and E on BC. Let FGH be parallel to AB with G on BC and H on AC. Let , , , and be the radii of the circumcircles , , , of the triangles ABC, DBE, EFG, and HGC, respectively. Then , with the plus sign when F is outside of ABC and minus when F is inside.
The statement of the theorem is in Problem 94. Similar Triangles, Circumradii, Parallel.