Problem

Source: Romanian National Olympiad, grade ix, p.3

Tags: geometry, median, area



Let be a point $ P $ in the interior of a triangle $ ABC. $ The lines $ AP,BP,CP $ meet $ BC,AC, $ respectively, $ AB $ at $ A_1,B_1, $ respectively, $ C_1. $ If $$ \mathcal{A}_{PBA_1} +\mathcal{A}_{PCB_1} +\mathcal{A}_{PAC_1} =\frac{1}{2}\mathcal{A}_{ABC} , $$show that $ P $ lies on a median of $ ABC. $ $ \mathcal{A} $ denotes area.