Sangaku for Incircles
The Sangaku (old Japanese theorem) states that the centers of the incircles of the four triangles defined by the sides and the diagonals inside any concyclic quadrilateral are vertices of a rectangle.
It is quite unexpected that the centers of the incircles form a rectangle, no matter how the concyclic quadrilateral is distorted. For a proof, decompose the angles of the figure made by the centers. Find relations with the angles of the concyclic quadrilateral. Use properties of the angles that are due to the quadrilateral being concyclic. The proof is not trivial.