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Characterize the convex quadrilateral

Source: Spanish Mathematical Olympiad 2013. Problem 6.

March 26, 2015
geometry

Problem Statement

Let ABCDABCD a convex quadrilateral where:
∣AB∣+∣CD∣=2∣AC∣|AB|+|CD|=\sqrt{2} |AC| and ∣BC∣+∣DA∣=2∣BD∣|BC|+|DA|=\sqrt{2} |BD|
What form does the quadrilateral have?
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