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Construction of a trapezoid

Source: Czech and Slovak Olympiad 1957, National Round, Problem 4

April 10, 2020
geometryconstructive geometrytrapezoidnational olympiad

Problem Statement

Consider a non-zero convex angle ∠POQ\angle POQ and its inner point MM. Moreover, let m>0m>0 be given. Construct a trapezoid ABCDABCD satisfying the following conditions: (1) vertices A,DA, D lie on ray OPOP and vertices B,CB,C lie on ray OQOQ, (2) diagonals ACAC and BDBD intersect in MM, (3) AB=mAB=m. Prove that your construction is correct and discuss conditions of solvability.
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