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equal sum of areas under homothety of a square (2010 Kyiv City MO 10.3)

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July 13, 2020
geometryhomothetysquareareas

Problem Statement

A point OO is chosen inside the square ABCDABCD. The square A′B′C′D′A'B'C'D' is the image of the square ABCDABCD under the homothety with center at point OO and coefficient k>1k> 1 (points A′,B′,C′,D′A', B', C', D' are images of points A,B,C,DA, B, C, D respectively). Prove that the sum of the areas of the quadrilaterals A′ABB′A'ABB' and C′CDD′C'CDD' is equal to the sum of the areas quadrilaterals B′BCC′B'BCC' and D′DAA′D'DAA'.
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