### MEMO 2017 ekipno problem 5

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12. rujna 2018.

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.

Izvor: Srednjoeuropska matematička olimpijada 2017, ekipno natjecanje, problem 5