Question 4
Let an acute triangle $ABC$ be inscribed in a circle $(O)$, with $AB\lt AC$. Let $I$ be the midpoint of the arc $BC$ not containing $A$. Choose $K\ne C$ on segment $AC$ such that $IK=IC$. Line $BK$ meets $(O)$ again at $D\ne B$ and meets line $AI$ at $E$. Line $DI$ meets line $AC$ at $F$.
(a) Prove $EF=BC/2$.
(b) Choose $M$ on line $DI$ so that $CM\parallel AD$. Line $KM$ meets line $BC$ at $N$. The circumcircle of $BKN$ meets $(O)$ again at $P\ne B$. Prove that line $PK$ passes through the midpoint of $AD$.
(a) Prove $EF=BC/2$.
(b) Choose $M$ on line $DI$ so that $CM\parallel AD$. Line $KM$ meets line $BC$ at $N$. The circumcircle of $BKN$ meets $(O)$ again at $P\ne B$. Prove that line $PK$ passes through the midpoint of $AD$.