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Square and Product of Lengths

Source: 2009 IrMO Paper 1 Problem 2

January 30, 2018
geometry

Problem Statement

Let ABCDABCD be a square. The line segment ABAB is divided internally at HH so that ∣AB∣⋅∣BH∣=∣AH∣2|AB|\cdot |BH|=|AH|^2. Let EE be the midpoints of ADAD and XX be the midpoint of AHAH. Let YY be a point on EBEB such that XYXY is perpendicular to BEBE. Prove that ∣XY∣=∣XH∣|XY|=|XH|.
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