IZhO 2013, Олимпиада по математика · първи ден — Задача 1

IX International Zhautykov Olympiad in Mathematics, Almaty, 2013, 15 January, 2013, 9.00–13.30, First day · 15 януари 2013 г. · 7 т.

Условие

Given a trapezoid $ABCD$ ($AD \parallel BC$) with $\angle ABC > 90^{\circ}$. Point $M$ is chosen on the lateral side $AB$. Let $O_1$ and $O_2$ be the circumcenters of the triangles $MAD$ and $MBC$ respectively. The circumcircles of the triangles $MO_1D$ and $MO_2C$ meet again at the point $N$. Prove that the line $O_1O_2$ passes through the point $N$.

Решение

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Оригинал в Архива: IZhO_2013_1day- mathem.doc