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BC - AM = CA - BM = AB - CM, A-excircle related

Source: II Soros Olympiad 1995-96 R1 11.9 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

June 3, 2024
excirclegeometry

Problem Statement

In triangle ABCABC, the side BC=aBC = a and the radius rr of the circle tangent to the side BC and the extensions of ABAB and ACAC (AA-excircle) are known. It is also known that inside the triangle there is a point MM such that BC−AM=CA−BM=AB−CMBC - AM = CA - BM = AB - CM Find the radius of the circle inscribed in the triangle BMCBMC.
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