Question 52
Let $ABC$ be a triangle with circumcircle $\Gamma$ and incentre $I$. Let $M$ be the midpoint of side $BC$. Denote by $D$ the foot of perpendicular from $I$ to side $BC$. The line through $I$ perpendicular to $AI$ meets sides $AB$ and $AC$ at $F$ and $E$ respectively. Suppose the circumcircle of triangle $AEF$ intersects $\Gamma$ at a point $X$ other than $A$. Prove that lines $XD$ and $AM$ meet on $\Gamma$.