Consider triangle ABC with D on AB such that AD : DB = 3 : 4, and DE drawn parallel to BC meeting AC at E. Separately, F is the point on AC with AF : FC = 3 : 4, and the line through F parallel to BC meets AB at G. What is the relationship between the two segments DE and GF?

By BPT, DE parallel to BC forces AE : EC = AD : DB = 3 : 4, so E is precisely the point that divides AC in the ratio 3 : 4 — which is how F was defined. A point dividing a segment in a given ratio is unique, so E and F coincide. Running the same argument on AB shows G coincides with D. Hence DE and GF are one and the same segment; they cannot be distinct, equal-length, or half-length copies.