Dividing a Triangle by Lines Parallel to Two Sides
Let ABC be a triangle and E a point on AC. Let D be on AB such that DE is parallel to BC and F be on BC such that EF is parallel to AB. Let , , and be the areas of ADE, EFC, and DEFB, respectively. Then .
See problem 1.33 in V. Prasolov, Problems in Plane and Solid Geometry, Vol. 1, Plane Geometry [PDF], (D. Leites, ed. and trans.).