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Right-angled trapezoid ABCD

Source: Sharygin First Round 2013, Problem 14

April 4, 2013
geometrytrapezoidcircumcircleanalytic geometryquadraticsAsymptotetrigonometry

Problem Statement

Let MM, NN be the midpoints of diagonals ACAC, BDBD of a right-angled trapezoid ABCDABCD (∡A=∡D=90∘\measuredangle A=\measuredangle D = 90^\circ). The circumcircles of triangles ABNABN, CDMCDM meet the line BCBC in the points QQ, RR. Prove that the distances from QQ, RR to the midpoint of MNMN are equal.
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