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Collinearity in isosceles trapezoid

Source: Nordic Mathematical Contest 2017 Problem 3

April 3, 2017
geometry

Problem Statement

Let MM and NN be the midpoints of the sides ACAC and ABAB, respectively, of an acute triangle ABCABC, AB≠ACAB \neq AC. Let ωB\omega_B be the circle centered at MM passing through BB, and let ωC\omega_C be the circle centered at NN passing through CC. Let the point DD be such that ABCDABCD is an isosceles trapezoid with ADAD parallel to BCBC. Assume that ωB\omega_B and ωC\omega_C intersect in two distinct points PP and QQ. Show that DD lies on the line PQPQ.
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