Problem

Source: ELMO Shortlist 2012, G1; also ELMO #1

Tags: geometry, circumcircle, power of a point, radical axis, cyclic quadrilateral, Elmo, Hi



In acute triangle $ABC$, let $D,E,F$ denote the feet of the altitudes from $A,B,C$, respectively, and let $\omega$ be the circumcircle of $\triangle AEF$. Let $\omega_1$ and $\omega_2$ be the circles through $D$ tangent to $\omega$ at $E$ and $F$, respectively. Show that $\omega_1$ and $\omega_2$ meet at a point $P$ on $BC$ other than $D$. Ray Li.