In given figures, sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ, QR and median PM of triangle PQR. Show that triangle ABC is similar to triangle PQR.

Given AD and PM are medians of triangle ABC and triangle PQR, ∴ BD = 1/2 BC and QM = 1/2 QR (1) Given that, AB/PQ = BC/QR = AD/PM (2) ∴ From (1) and (2), AB/PQ = BD/QM = AD/PM.. (3) In triangle ABD and triangle PQM AB/PQ = BD/QM = AD/PM [From (3)] ∴ By SSS criterian of proportionality triangle ABD is similar to triangle PQM ∴ ∠B = ∠Q (Corresponding Sides of Similar Triangles) (4) In triangle ABC and triangle PQR AB/PQ = BC/QR (From 2) ∠B = ∠Q (From 4) ∴ By SAS criterian of proportionality triangle ABC is similar to triangle PQR [hence proved]