MEMO 2017 Problem 5

Let $ABC$ be an acute-angled triangle with $AB > AC$ and circumcircle $\Gamma$ . Let $M$ be the midpoint of the shorter arc $BC$ of $\Gamma$ , and let $D$ be the intersection of the rays $AC$ and $BM$ . Let $E \neq C$ be the intersection of the internal bisector of the angle $ACB$ and the circumcircle of the triangle $BDC$ . Let us assume that $E$ is inside the triangle $ABC$ and there is an intersection $N$ of the line $DE$ and the circle $\Gamma$ such that $E$ is the midpoint of the segment $DN$ . Show that $N$ is the midpoint of the segment $I_B I_C$ , where $I_B$ and $I_C$ are the excentres of $ABC$ opposite to $B$ and $C$ , respectively.