D is a point on side BC of triangle ABC with BD : DC = 1 : 2, and E is the midpoint of AD. The line BE, extended, meets AC at F. Find AF : FC.

Mass points: BD : DC = 1 : 2 gives masses B = 2, C = 1, so D = 3. E is the midpoint of AD, so A also has mass 3. On AC, AF : FC = mass C : mass A = 1 : 3. (Check with A(0, 3), B(0, 0), C(3, 0): F = (0.75, 2.25), a quarter of the way from A.)