Two Triangles of Equal Area on Either Side of an Angle Bisector
Let ABC be a triangle and let the angle bisector at vertex C intersect the circumcircle at R and the perpendicular bisectors of BC and AC at P and Q, respectively. Let the midpoints of BC and AC be S and T, respectively. Then RQT and RPS have equal area.
See IMO 2007 shortlist geometry problem G1.