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GeoGebra Team Competitions - First round, Problem 3.

Different solutions

Given a triangle ABC, take an internal point O in it. Consider the 3 parallels to the sides respectively passing through O, and intersecting the sides of the triangle. Denote by R_1, R_2, R_3 the radii of the circles around the 3 smaller triangles determined by one vertex of the triangle and such a parallel line segments, and denote by R the Radius of the circumcircle of ABC. Prove that: R_1+R_2+R_3=R.