Question 5
Circle $\left(O\right)$ has diameter $AB$ and $M$ is the midpoint of $OB$. The perpendicular to $AB$ at $M$ meets the circle at $C$. Choose $E$ inside $AB$; $CE$ meets the circle again at $D$. Let $BH \perp CD , H \in CD$. Prove $CH = \frac{AD}{2}$ and that the midpoint I of $DF$, where $BH$ meets the circle again at $F$, lies on a fixed circle.