1989 OIM Problems/Problem 4

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Problem

The circumference inscribed in triangle $ABC$, is tangent to sides $AB$ and $AC$ at points $M$ and $N$ respectively. The bisectors of $A$ and $B$ intersect $MN$ at points $P$ and $Q$ respectively. Let $O$ be the incenter of triangle $ABC$. Prove: \[(MP)(OA)=(BC)(OQ)\]

~translated into English by Tomas Diaz. ~orders@tomasdiaz.com

Solution

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See also

https://www.oma.org.ar/enunciados/ibe4.htm