Given: The theorem: if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points , then the other two sides are divided in the same ratio. To find: Prove the theorem (BASIC Proportionality Theorem) Solution: First we need to do some constructions. In triangle PQR, LET l be a line parallel to QR, let the INTERSECTED points by l be M on PQ and N on PR, and now join QN and MR. Now After looking to the figure, we get:                area( tri MPN ) / area ( tri NQM ) = PM / MQ ( i ) As both triangles have the height same (MN) and a common vertex (M). Similarly :           area( tri MPN ) / area ( tri NRM ) = PN / NR  (ii) Now:           area( tri NQM ) = area ( tri NRM ) iii)           (triangle LIES in parallel lines and have same base) Now from equation i, ii and iii, we have:           area( tri MPN )/  area ( tri NQM ) = area( tri MPN ) = area ( tri NRM ) So similarly:           PM / MQ =  PN / NR           Hence proved.