Let ABCD be a cyclic quadrilateral with circumcenter O. Let the internal angle bisectors at A and B meet at X, the internal angle bisectors at B and C meet at Y, the internal angle bisectors at C and D meet at Z, and the internal angle bisectors at D and A meet at W. Further, let AC and BD meet at P. Suppose that the points X, Y, Z, W, O, and P are distinct.
Prove that O, X, Y, Z, W lie on the same circle if and only if P, X, Y, Z, and W lie on the same circle. EGMOgeometrycirclesEGMO2022