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<BOP =<AOR wanted, trapezoid ABCD, circumcircles of AMC and BND

Source: 2021 Sharygin Geometry Olympiad Finals grade IX p6

August 2, 2021
geometrytrapezoidequal angles

Problem Statement

The diagonals of trapezoid ABCDABCD (BC∥ADBC\parallel AD) meet at point OO. Points MM and NN lie on the segments BCBC and ADAD respectively. The tangent to the circle AMCAMC at CC meets the ray NBNB at point PP; the tangent to the circle BNDBND at DD meets the ray MAMA at point RR. Prove that ∠BOP=∠AOR\angle BOP =\angle AOR.
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