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3 incircles inside a trapezoid, angle bisectors of A,D intersec on side BC

Source: Oral Moscow Geometry Olympiad 2015 grades 8-9 p4

August 8, 2019
geometrytrapezoidangle bisectorincircleequal segments

Problem Statement

In trapezoid ABCDABCD, the bisectors of angles AA and DD intersect at point EE lying on the side of BCBC. These bisectors divide the trapezoid into three triangles into which the circles are inscribed. One of these circles touches the base ABAB at the point KK, and two others touch the bisector DEDE at points MM and NN. Prove that BK=MNBK = MN.
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