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distance between touchpoints of side with incircle and excircle

Source: 2009 Sharygin Geometry Olympiad Final Round problem 8 grade 8

July 26, 2018
geometryincircleexcircle

Problem Statement

A triangle ABCABC is given, in which the segment BCBC touches the incircle and the corresponding excircle in points MM and NN. If ∠BAC=2∠MAN\angle BAC = 2 \angle MAN, show that BC=2MNBC = 2MN.
(N.Beluhov)
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