Romantics of Geometry, Pr. 247 FJGC, https://www.facebook.com/103907057666827/photos/a.103973994326800/613689746688553

 Problem 247, FJGC
Let O be the circumcenter of triangle ABC and A' reflection of A about BO. AA' intersects BC at U. Prove that C and U are inverses of each other w.r.t. circle (B,BA)

Proof:

BC is angle bisector of <ACA', thus <A'CB=<ACB=<BAA', so AB is tangent to the circle (AUC), giving AB.AB=BU.BC, done.