MathNet 2016-part-06 — 184 problems (EN–VI) Question 183

The bisector of an angle $BAC$ of acute angled triangle $ABC$ ($AC \neq AB$) intersects its circumcircle second time at the point $W$. Let $O$ be the circumcenter of $\triangle ABC$. Line $AW$ intersects second time the circumcircles of triangles $OWB$ and $OWC$ at points $N$ and $M$ respectively. Prove that $BN + MC = AW$.

(V.Mitrofanov, D.Khilko)