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Three trapezoids with inscribed circle

Source: IV Caucasus Mathematic Olympiad

April 7, 2019
geometryKvant

Problem Statement

On sides BCBC, CACA, ABAB of a triangle ABCABC points KK, LL, MM are chosen, respectively, and a point PP is inside ABCABC is chosen so that PL∥BCPL\parallel BC, PM∥CAPM\parallel CA, PK∥ABPK\parallel AB. Determine if it is possible that each of three trapezoids AMPLAMPL, BKPMBKPM, CLPKCLPK has an inscribed circle.
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