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Tournament of the Towns SP2010 (I guess), Trapezoid

Source:

May 8, 2010
geometrytrapezoidratiogeometric transformationreflectionincenteranalytic geometry

Problem Statement

Quadrilateral ABCDABCD is circumscribed around the circle with centre II. Let points MM and NN be the midpoints of sides ABAB and CDCD respectively and let IMAB=INCD\frac{IM}{AB} = \frac{IN}{CD}. Prove that ABCDABCD is either a trapezoid or a parallelogram.
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