Subtriangles Formed by Concurrent Lines Parallel to the Sides of a Triangle
Let ABC be a triangle and P be an interior point. Draw lines through P parallel to the sides of ABC that intersect AB at C' and C'', BC at A' and A'', and CA at B' and B'', with A'B'' parallel to AB, B'C'' parallel to BC, and C'A'' parallel to CA. Let , , , and be the areas of ABC, PA'A'', PB'B'', and PC'C'', respectively. Then .
See problem 1.35 in V. Prasolov, Problems in Plane and Solid Geometry, Vol. 1, Plane Geometry [PDF], (D. Leites, ed. and trans.).