In triangle ABC, the point D lies on BC with BD : DC = 1 : 2, and E is the midpoint of AD. The ray BE is extended to meet AC at F. What is the ratio AF : FC?

Use mass points. To get BD : DC = 1 : 2, give B a mass of 2 and C a mass of 1, so D carries mass 3. Since E is the midpoint of AD, A must carry the same mass as D, namely 3. On the line through A and C, the point F divides AC in the inverse ratio of the masses at its ends: AF : FC = (mass at C) : (mass at A) = 1 : 3. Placing B at (0, 0), C at (3, 0) and A at (0, 3) confirms it — F comes out at (0.75, 2.25), one quarter of the way from A to C.