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## Sum independent of triangulation

In a circle with center $O$ is inscribed a polygon which is triangulated. Show that the sum of the squares of the distances from $O$ to the incenters of the formed triangles is independent of the triangulation.
Romanian TST 2007

Hint: All you need is to prove this fact for the cyclic quardilateral, which has exactly two triangulations. Then you can solve the real problem by induction over the numbers of vertices of the polygon.