VTMO 2014 — Grade 9 — final Question 4

Circle $\left(O\right)$ has diameter $AB$. Let $C , D$ lie on it with $CD \parallel AB$. Choose $E$ on $AB$ with $AE = BD$. Extend $CE$ to meet the circle at $M$, and $DO$ to meet it at $N$. Let $F = AN \cap BM$. Prove $EF \perp AB$.