DE-MO 2005 Problem 5

(a) [Problem for class 11]

Let r be the inradius and $r_a$ , $r_b$ , $r_c$ the exradii of a triangle ABC. Prove that $\frac{1}{r}=\frac{1}{r_a}+\frac{1}{r_b}+\frac{1}{r_c}$ .

(b) [Problem for classes 12/13]

Let r be the radius of the insphere and let $r_a$ , $r_b$ , $r_c$ , $r_d$ 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 $\frac{2}{r}=\frac{1}{r_a}+\frac{1}{r_b}+\frac{1}{r_c}+\frac{1}{r_d}$ .

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