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The incircle of the triangle $A B C$ is tangent to sides $A B$ and $A C$ at $M$ and $N$, respectively. The bisectors of the angles at $A$ and $B$ intersect $M N$ at points $P$ and $Q$, respectively. Let $O$ be the incenter of $\triangle A B C$. Prove that $M P \cdot O A=B C \cdot O Q$.
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