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midpoint of MN lies on the circumcircle of PQR, starting with isosceles

Source: Ukraine TST 2013 p1

April 28, 2020
midpointgeometryisoscelescircumcircle

Problem Statement

Let ABCABC be an isosceles triangle ABCABC with base BCBC insribed in a circle. The segment ADAD is the diameter of the circle, and point PP lies on the smaller arc BDBD. Line DPDP intersects rays ABAB and ACAC at points MM and NN, and the lines BPBP and CPCP intersects the line ADAD at points QQ and RR. Prove that the midpoint of the segment MNMN lies on the circumscribed circle of triangle PQRPQR.
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