Question 35
Let $AB\Gamma$ be an acute angled triangle with $AB \lt A\Gamma \lt B\Gamma$ and let $c$ be its circumcircle with centre $O$. At the small arcs $A\Gamma$ and $AB$ we consider the points $\Delta$ and $E$ respectively. Let $K$ be the intersection point of $B\Delta$, $\Gamma E$ and $N$ be the second common point of the circumcircles of the triangles $BKE$, let it be $c_1$, and $\Gamma K\Delta$, let it be $c_2$. Prove that the points $A, K, N$ are collinear if and only if the point $K$ lies on the A-symmedian of the triangle $AB\Gamma$.