Question 101
$D$, $E$, $F$ are points on the sides $AB$, $BC$, $CA$, respectively, of a triangle $ABC$ such that $AD = AF$, $BD = BE$ and $DE = DF$. Let $I$ be the incircle of the triangle $ABC$, and let $K$ be the point of intersection of the line $BI$ and the tangent line through $A$ to the circumcircle of the triangle $ABI$. Show that $AK = EK$ if $AK = AD$.