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find chord of one circle tangent to another one, incircle related

Source: Sharygin 2005 finals 11.4

August 31, 2019
geometryangleincirclechorddiameter

Problem Statement

In the triangle ABC,A=α,BC=aABC , \angle A = \alpha, BC = a. The inscribed circle touches the lines ABAB and ACAC at points MM and PP. Find the length of the chord cut by the line MPMP in a circle with diameter BCBC.