Proof of converse of BPT theorum

Converse of Basic Proportionality TheoremStatement If a line divides any two sides of a triangle (Δ) in the same ratio, then the line must be parallel (||) to the third side.DiagramGiven In ΔABC, D and E are the two points of AB and AC respectively, such that, AD/DB = AE/EC.To Prove DE || BCProof In ΔABC,given, AD/DB = AE/EC ----- (1) Let us assume that in ΔABC, the point F is an intersect on the side AC. So we can apply the Thales Theorem, AD/DB = AF/FC ----- (2) Simplify, in (1) and (2) == AE/EC = AF/FC Add 1 on both sides, == (AE/EC) + 1 = (AF/FC) + 1 == (AE+EC)/EC = (AF+FC)/FC == AC/EC = AC/FC == EC = FC From the above, we can say that the points E and F coincide on AC. i.e., DF coincides with DE. Since DF is parallel to BC, DE is also parallel BC Hence theConverse of Basic Proportionality theroremis proved.