another proof

1) draw 3 additional lines parallel to each cevian passing through the remaining vertices of 2 triangles.

2) show that 6 other shaded triangles are mirror images of the green triangle and therefore have the same area

3) show that every part of a shaded triangle that lies outside of the big triangle has an equal unshaded part inside the big triangle. Midpoints of big triangle's sides are points of symmetry.