Team Selection Test

Question

Let a, b, c be the lengths of the sides of a triangle; r_a, r_b, r_c be the corresponding exradii, respectively, and r be the inradius. Prove that a+b+c2a^2+b^2+c^2 r_a^2+r_b^2+r_c^2r_a+r_b+r_c-3r

Accepted Answer

Note that a+b+c2r = b+c-a2r_a = a+b+c2r_a - ar_a = a+b+c2r_b - br_b = a+b+c2r_c - cr_c. Therefore, we have 3r(a+b+c)2 = a+b+c2(r_a + r_b + r_c) - ar_a - br_b - cr_c and hence ar_a + br_b + cr_c = a+b+c2(r_a + r_b + r_c - 3r). On the other hand, the Cauchy-Schwarz inequality implies ar_a + br_b + cr_c a^2 + b^2 + c^2 r_a^2 + r_b^2 + r_c^2 and the result follows.