Question 108
Let $ABC$ be an isosceles triangle with apex $A$ and $AB \gt BC$. Let $k$ be the circle with centre $A$ through $B$ and $C$. Let $H$ be the second intersection of $k$ with the altitude of triangle $ABC$ through $B$. Further, let $G$ be the second intersection of $k$ with the median through $B$ in triangle $ABC$. Let $X$ be the intersection of the lines $AC$ and $GH$. Show that $C$ is the midpoint of the segment $AX$.