IT-CESENATICO 2014 Problem 4

Let $\omega$ be a circle with center $A$ and radius $R$ . On the circumference of $\omega$ four distinct points $B, C, G, H$ are taken in that order in such a way that $G$ lies on the extended $B$ -median of the triangle $ABC$ , and H lies on the extension of altitude of $ABC$ from $B$ . Let $X$ be the intersection of the straight lines $AC$ and $GH$ . Show that the segment $AX$ has length $2R$ .