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ASU 170 All Soviet Union MO interior point isosceles triangles with all vertices

Source:

July 3, 2019
isoscelesgeometry

Problem Statement

The point OO inside the convex polygon makes isosceles triangle with all the pairs of its vertices. Prove that OO is the centre of the circumscribed circle.
other formulation: PP is a convex polygon and XX is an interior point such that for every pair of vertices A,BA, B, the triangle XABXAB is isosceles. Prove that all the vertices of PP lie on a circle with center XX.