Infographic Geometry Problem 1067: Acute Triangle, Orthocenter, Circumradius, Inradius, Exradii, Distance, Diameter. Level: High School, SAT Prep, College, Mathematics Education

The figure shows an acute triangle ABC with the orthocenter H, the circumradius R, the inradius r, and the exradii, r_a, r_b, and r_c.
If AH = a₁, BH = b₁, and CH = c₁, prove that r_a + r_b + r_c + r = + a₁ + b₁ + c₁ + 2R.