In the figure, A, B and C are points on OP, OQ and OR respectively such that AB || PQ and AC || PR. Show that BC || QR.

Solution :

Given : O  is any point within triangle PQR, AB || PQ and AC || PR

To Prove :  BC || QR

Construction : Join BC

Proof : In triangle OPQ,

Given,      AB || PQ

By basic proportionality theorem,

\(OA\over AP\) = \(OB\over BQ\)                  ………..(1)

In triangle OPR,

Given,        AC || PR

By basic proportionality theorem,

\(OA\over AP\) = \(OC\over CR\)                 ………..(2)

From (1) and (2), we obtain that

\(OB\over BQ\) = \(OC\over CR\)

Hence, by converse of basic proportionality theorem, we have

BC || QR