IMO 1967 LL BUL4

Origin: BUL

Problem

Suppose medians ma and mb of a triangle are orthogonal. Prove that: (a) The medians of that triangle correspond to the sides of a right-angled triangle. (b) The inequality 5(a2 + b2 −c2) \geq8ab is valid, where a, b, and c are side lengths of the given triangle.

Solution

(a) Let ABCD be a parallelogram, and K, L the midpoints of segments BC and CD respectively. The sides of \triangleAKL are equal and parallel to the medians of \triangleABC. (b) Using the formulas 4m2 a = 2b2 + 2c2 −a2 etc., it is easy to obtain that m2 a + m2 b = m2 c is equivalent to a2 + b2 = 5c2. Then 5(a2 + b2 −c2) = 4(a2 + b2) \geq8ab.