Question 45
Let $ABC$ be a triangle with $AB \lt AC$ and let $D$ be the other intersection point of the angle bisector of $A$ with the circumcircle of triangle $ABC$. Let $E$ and $F$ be points on the sides $AB$ and $AC$ respectively, such that $AE = AF$ and let $P$ be the point of intersection of $AD$ and $EF$. Let $M$ be the midpoint of $BC$. Prove that $AM$ and the circumcircles of triangles $AEF$ and $PMD$ pass through a common point.