Problem

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Tags: geometry, circumcircle, geometry proposed



Let $O$ be the circumcenter,$R$ be the circumradius, and $k$ be the circumcircle of a triangle $ABC$ . Let $k_1$ be a circle tangent to the rays $AB$ and $AC$, and also internally tangent to $k$. Let $k_2$ be a circle tangent to the rays $AB$ and $AC$ , and also externally tangent to $k$. Let $A_1$ and $A_2$ denote the respective centers of $k_1$ and $k_2$. Prove that: $(OA_1+OA_2)^2-A_1A_2^2 = 4R^2.$