Problem

Source: 2022 China TST, Test 1, P4

Tags: geometry, incenter



Let $ABC$ be an acute triangle with $\angle ACB>2 \angle ABC$. Let $I$ be the incenter of $ABC$, $K$ is the reflection of $I$ in line $BC$. Let line $BA$ and $KC$ intersect at $D$. The line through $B$ parallel to $CI$ intersects the minor arc $BC$ on the circumcircle of $ABC$ at $E(E \neq B)$. The line through $A$ parallel to $BC$ intersects the line $BE$ at $F$. Prove that if $BF=CE$, then $FK=AD$.