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USS cot triangle equality [equal-inradius cevian length]

Source: IMO ShortList 1988, Problem 30, USS 1, Problem 84 of ILL

November 9, 2005
geometryinradiustrigonometryarea of a triangleIMO Shortlist

Problem Statement

A point M M is chosen on the side AC AC of the triangle ABC ABC in such a way that the radii of the circles inscribed in the triangles ABM ABM and BMC BMC are equal. Prove that BM^{2} \equal{} X \cot \left( \frac {B}{2}\right) where X is the area of triangle ABC. ABC.
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