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Prove that the line O1O2 passes through the point N

Source: International Zhautykov Olympiad 2013 - D1 - P1

January 17, 2013
geometrytrapezoidcircumcirclegeometric transformationgeometry unsolved

Problem Statement

Given a trapezoid ABCDABCD (AD∥BCAD \parallel BC) with ∠ABC>90∘\angle ABC > 90^\circ . Point MM is chosen on the lateral side ABAB. Let O1O_1 and O2O_2 be the circumcenters of the triangles MADMAD and MBCMBC, respectively. The circumcircles of the triangles MO1DMO_1D and MO2CMO_2C meet again at the point NN. Prove that the line O1O2O_1O_2 passes through the point NN.
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