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heights and inner radius of a triangle (BWM1988_1_2)

Source: Bundeswettbewerb Mathematik 1988, first stage, second question

September 6, 2003
geometryinradiusgeometry solved

Problem Statement

Let hah_a, hbh_b and hch_c be the heights and rr the inradius of a triangle. Prove that the triangle is equilateral if and only if ha+hb+hc=9rh_a + h_b + h_c = 9r.