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3 triangles with same inradii given, r_1 +r_2 =r wanted

Source: Polish second round 1993 p5

January 19, 2020
geometryinradiusSumincircle

Problem Statement

Let D,E,FD,E,F be points on the sides BC,CA,ABBC,CA,AB of a triangle ABCABC, respectively. Suppose that the inradii of the triangles AEF,BFD,CDEAEF,BFD,CDE are all equal to r1r_1. If r2r_2 and rr are the inradii of triangles DEFDEF and ABCABC respectively, prove that r1+r2=rr_1 +r_2 =r.
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