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isosceles wanted, AD//BC. 2 circumcircles related

Source: 2016 VWO Flanders MO p1

August 7, 2020
geometrycircumcircleisoscelesparallel

Problem Statement

In the quadrilateral ABCDABCD is AD∥BCAD \parallel BC and the angles ∠A\angle A and ∠D\angle D are acute. The diagonals intersect in PP. The circumscribed circles of △ABP\vartriangle ABP and △CDP\vartriangle CDP intersect the line ADAD again at SS and TT respectively. Call MM the midpoint of [ST][ST]. Prove that △BCM\vartriangle BCM is isosceles. https://1.bp.blogspot.com/-C5MqC0RTqwY/Xy1fAavi_aI/AAAAAAAAMSM/2MXMlwb13McCYTrOHm1ZzWc0nkaR1J6zQCLcBGAsYHQ/s0/flanders%2B2016%2Bp1.png
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