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maximal radii of circles inscribed into 2 crescents are equal

Source: 2012 Sharygin Geometry Olympiad Final Round 10.5

August 3, 2018
geometryarccirclesinscribed circles

Problem Statement

A quadrilateral ABCDABCD with perpendicular diagonals is inscribed into a circle ω\omega. Two arcs α\alpha and β\beta with diameters AB and CDCD lie outside ω\omega. Consider two crescents formed by the circle ω\omega and the arcs α\alpha and β\beta (see Figure). Prove that the maximal radii of the circles inscribed into these crescents are equal.
(F.Nilov)
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