# Equal Segments

 A problem from the geometer Jean-Louis Ayme, see http://www.artofproblemsolving.com/Forum/viewtopic.php?f=47&p=3590641. Let $ABC$be a triangle, with circumcenter $O$. Denote the intersection of the internal bisector of $\angle BAC$ with $BC$ as $M$. Let $P$ be on $CA$such that $MP\perp AO$. Prove that $AB = AP$.

# Perpendicular bisector

 A surprising result from the 36th Spanish Mathematical Olympiad, see solution http://www.artofproblemsolving.com/Forum/blog.php?u=214539&b=107527. Let $\mathcal{S}_1, \mathcal{S}_2$ be two circles of different radii, centers $O_1, O_2$ respectively, which intersect at $A,B$. Let $PQ$ be a line through $B$ meeting $\mathcal{S}_1, \mathcal{S}_2$ at $P,Q$ respectively. Prove that a) For any choice of $\overline{PQ}$, the perpendicular bisector of $PQ$ passes through a fixed point $M$, and b) $AO_1O_2M$ is an isosceles trapezium.

# APMO 2014 Q5

 APMO 2014 Q5, see solution http://www.artofproblemsolving.com/Forum/blog.php?u=214539&b=109581. Circles $\omega$ and $\Omega$ meet at points $A$ and $B$. Let $M$ be the midpoint of the arc $AB$ of circle $\omega$ ($M$ lies inside $\Omega$). A chord $MP$ of circle $\omega$ intersects $\Omega$ at $Q$ ($Q$ lies inside $\omega$). Let $\ell_P$ be the tangent line to $\omega$ at $P$, and let $\ell_Q$ be the tangent line to $\Omega$ at $Q$. Prove that the circumcircle of the triangle formed by the lines $\ell_P$, $\ell_Q$ and $AB$ is tangent to $\Omega$.