In triangle ABC, D is the midpoint of AB. A line through D drawn parallel to BC meets AC at E, and a line through E drawn parallel to AB meets BC at F. If BC = 18 cm, what is the length of BF?

By the converse of the mid-point theorem, a line through the midpoint of one side parallel to a second side bisects the third side. So E is the midpoint of AC. Applying the same converse in triangle ABC to the line EF (through the midpoint E of AC, parallel to AB) shows F is the midpoint of BC. Hence BF = 18/2 = 9 cm.