Question 1
Let $(O)$ be a circle with diameter $AB$ and let $C$ lie on the circle. The line through $C$ parallel to $AB$ meets the circle at $D$. Choose $E$ on $AB$ with $AE=BD$. Extend $CE$ to meet the circle at $M$, and extend $DO$ to meet it at $N$. Let $F=AN\cap BM$. Prove that $EF\perp AB$.