Concyclic Points Associated with an Angle Bisector and an Excircle
Let ABC be a triangle with incenter I and circumcenter O, and let J be the excenter opposite A. Let the angle bisector from A intersect the circumcircle at M. Then B, C, I, and J lie on a circle centered at M.
See problem 2.4(a) in V. Prasolov, Problems in Plane and Solid Geometry, Vol. 1, Plane Geometry [PDF], (D. Leites, ed. and trans.).