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trapezoid, prove two equations

Source: National round in Slovakia 2004/2005, problem 3; contest 1996

April 10, 2005
geometrytrapezoidratiorectangleparallelogramgeometric transformationhomothety

Problem Statement

Let ABCDABCD be a trapezoid such that ABAB is parallel to CDCD, and let EE be the midpoint of its side BCBC. Suppose we can inscribe a circle into the quadrilateral ABEDABED, and that we can inscribe a circle into the quadrilateral AECDAECD. Denote ∣AB∣=a|AB|=a, ∣BC∣=b|BC|=b, ∣CD∣=c|CD|=c, ∣DA∣=d|DA|=d. Prove that a+c=b3+d;a+c=\frac{b}{3}+d; 1a+1c=3b\frac{1}{a}+\frac{1}{c}=\frac{3}{b}
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