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AC // BM wanted, right triangle, projections, midpoints, circumcenter

Source: 2017 Grand Duchy of Lithuania, Mathematical Contest p3 (Baltic Way TST)

October 3, 2020
geometryparallelright trianglecircumcircle

Problem Statement

Let ABCABC be a triangle with ∠A=90o\angle A = 90^o and let DD be an orthogonal projection of AA onto BCBC. The midpoints of ADAD and ACAC are called EE and FF, respectively. Let MM be the circumcentre of △BEF\vartriangle BEF. Prove that AC∥BMAC\parallel BM.
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