Question 1
Let $(O)$ be the circle with diameter $AB$, and let $M$ be the midpoint of $OB$. The line through $M$ perpendicular to $AB$ meets the circle at $C$. Choose $E$ on $AB$, distinct from $A,B$, and let $CE$ meet the circle again at $D$. Let $H$ be the foot of the perpendicular from $B$ to $CD$.
a) Prove $CH=\dfrac12AD$.
b) Let $BH$ meet the circle again at $F$, and let $I$ be the midpoint of $DF$. Prove that $I$ lies on a fixed circle as $E$ varies.
a) Prove $CH=\dfrac12AD$.
b) Let $BH$ meet the circle again at $F$, and let $I$ be the midpoint of $DF$. Prove that $I$ lies on a fixed circle as $E$ varies.