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Equivalence between a right angle and sum of two segments in a square

Source: KAMO 2022 Grades 7-8 P3

July 5, 2022
geometry

Problem Statement

Let ABCDABCD be a square and let MM be the midpoint of BCBC. Let XX and YY be points on the segments ABAB and CDCD, respectively. Prove that ∠XMY=90∘\angle XMY = 90^\circ if and only if BX+CY=XYBX + CY = XY.
Note: In the competition, students were only asked to prove the 'only if' direction.
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