Question 5
Given an acute, scalene triangle $ABC$ ($AB \lt AC$) with the circumcircle $(O)$. Let $I$ be the midpoint of arc $BC$ that does not contain $A$. Take $K$ on $AC$ ($K \neq C$) such that $IK = IC$. Line $BK$ intersects $(O)$ again at $D$ and intersects $AI$ at $E$. $DI$ meets $AC$ at $F$.
a) Prove that $EF = \frac{BC}{2}$.
b) Let $M$ be the point on $DI$ such that $CM$ is parallel to $AD$. Line $KM$ meets $BC$ at $N$ and the circumcircle of triangle $BKN$ meets $(O)$ again at $P$. Prove that $PK$ passes through the midpoint of the segment $AD$.
a) Prove that $EF = \frac{BC}{2}$.
b) Let $M$ be the point on $DI$ such that $CM$ is parallel to $AD$. Line $KM$ meets $BC$ at $N$ and the circumcircle of triangle $BKN$ meets $(O)$ again at $P$. Prove that $PK$ passes through the midpoint of the segment $AD$.