Problem

Source: ELMO 2014 Shortlist G12, by David Stoner

Tags: geometry, circumcircle, ratio, symmetry, projective geometry, geometry solved, trigonometry



Let $AB=AC$ in $\triangle ABC$, and let $D$ be a point on segment $AB$. The tangent at $D$ to the circumcircle $\omega$ of $BCD$ hits $AC$ at $E$. The other tangent from $E$ to $\omega$ touches it at $F$, and $G=BF \cap CD$, $H=AG \cap BC$. Prove that $BH=2HC$. Proposed by David Stoner