SAMC

Question

Let ABC be an acute triangle and let MNPQ be a square inscribed in the triangle such that M, N BC, P AC, Q AB. Prove that area[MNPQ] 12 area[ABC] FIGURE_0

Accepted Answer

FIGURE_0 Denote by x the length of sides of square MNPQ, a = BC, h_a = AA', where AA' BC. The triangle AQP and ABC are similar, hence we have xa = h_a - xh_a. We get x = a h_aa + h_a a h_a2 a h_a = 12 a h_a = 12 2 area[ABC] and the desired inequality follows. The equality holds if and only if h_a = a.