Problem

Source: Iran 3rd round 2011-geometry exam-p2

Tags: geometry, circumcircle, trapezoid, cyclic quadrilateral, perpendicular bisector, geometry proposed, Iran



In triangle $ABC$, $\omega$ is its circumcircle and $O$ is the center of this circle. Points $M$ and $N$ lie on sides $AB$ and $AC$ respectively. $\omega$ and the circumcircle of triangle $AMN$ intersect each other for the second time in $Q$. Let $P$ be the intersection point of $MN$ and $BC$. Prove that $PQ$ is tangent to $\omega$ iff $OM=ON$. proposed by Mr.Etesami