Question 20
Let $\omega$ be the circumcircle of a triangle $ABC$. Let $P$ be a point on the arc $AB$ of $\omega$, not containing $C$. The circle inscribed in triangle $ABC$ has center $I$, and the line $AI$ intersects $\omega$ at a point $M$, distinct from $A$. The circumcircle of triangle $API$ intersects segment $AB$ at a point $E$ (other than $A$), and segments $PM$ and $BC$ intersect at point $D$. Prove that $BE = BD$.