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tangent quadrilaterals ABED and AECD in trapezoid ABCD

Source: Czech and Slovak MO, III A, 2005 p3

January 12, 2020
geometrytrapezoidtangent

Problem Statement

In a trapezoid ABCDABCD with AB//CD,EAB // CD, E is the midpoint of BCBC. Prove that if the quadrilaterals ABEDABED and AECDAECD are tangent, then the sides a=AB,b=BC,c=CD,d=DAa = AB, b = BC, c =CD, d = DA of the trapezoid satisfy the equalities a+c=b3+da+c = \frac{b}{3} +d and 1a+1c=3b\frac1a +\frac1c = \frac3b .
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