Question 123
Let $ABC$ be a triangle, let $K$ be the foot of the angle bisector relative to $BC$, and let $J$ be the foot of the trisector relative to $BC$ closer to side $AC$ (that is, $J$ is the point on $BC$ such that $3 \cdot \angle CAJ = \angle CAB$). Let $C'$ and $B'$ be two points on line $AJ$, on the same side of $A$ as $J$, such that $AC' = AC$ and $AB = AB'$. Prove that the quadrilateral $ABB'C$ is cyclic if and only if the lines $C'K$ and $B'B$ are parallel.