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1 / r = 1 / r_a + 1 / r_b + 1 / r_c for tetrahedron

Source: DeMO 2005, classes 11 and 12/13, 2nd day, problem 5

May 10, 2005
geometry3D geometrytetrahedroninradiusspheregeometry proposed

Problem Statement

(a) [Problem for class 11] Let r be the inradius and rar_a, rbr_b, rcr_c the exradii of a triangle ABC. Prove that 1r=1ra+1rb+1rc\frac{1}{r}=\frac{1}{r_a}+\frac{1}{r_b}+\frac{1}{r_c}. (b) [Problem for classes 12/13] Let r be the radius of the insphere and let rar_a, rbr_b, rcr_c, rdr_d the radii of the four exspheres of a tetrahedron ABCD. (An exsphere of a tetrahedron is a sphere touching one sideface and the extensions of the three other sidefaces.) Prove that 2r=1ra+1rb+1rc+1rd\frac{2}{r}=\frac{1}{r_a}+\frac{1}{r_b}+\frac{1}{r_c}+\frac{1}{r_d}. I am really sorry for posting these, but else, Orl will probably post them. This time, we really did not have any challenging problem on the DeMO. But at least, the problems were simple enough that I solved all of them. ;) Darij
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