Problem

Source: Germany VAIMO 2015 - #3

Tags: geometry, circumcircle, Intersection



Let $ABC$ be an acute triangle with $|AB| \neq |AC|$ and the midpoints of segments $[AB]$ and $[AC]$ be $D$ resp. $E$. The circumcircles of the triangles $BCD$ and $BCE$ intersect the circumcircle of triangle $ADE$ in $P$ resp. $Q$ with $P \neq D$ and $Q \neq E$. Prove $|AP|=|AQ|$. (Notation: $|\cdot|$ denotes the length of a segment and $[\cdot]$ denotes the line segment.)