IMO 1969 LL BUL10

Origin: BUL

Problem

Let M be the point inside the right-angled triangle ABC (\angleC = 90◦) such that \angleMAB = \angleMBC = \angleMCA = ϕ. Let \psi be the acute angle between the medians of AC and BC. Prove that sin(ϕ+\psi) sin(ϕ−\psi) = 5.