USAMO 2003 — Days 1 and 2 Question 4

Let $ABC$ be a triangle. A circle passing through $A$ and $B$ intersects segments $AC$ and $BC$ at $D$ and $E$, respectively. Lines $AB$ and $DE$ intersect at $F$, while lines $BD$ and $CF$ intersect at $M$. Prove that $MF=MC$ if and only if $MB\cdot MD=MC^2$.

Proposed by Titu Andreescu and Zuming Feng (as credited in the supplied source).