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

We know that,

If a line is drawn parallel to one side of a triangle to intersect the other two sides at distinct points, the other two sides are divided in the same ratio.

In Δ OPQ,

AB || PQ [∵ Given]

$\therefore \dfrac{OA}{AP} = \dfrac{OB}{BQ}$ ………….. (1)

In Δ OPR,

AC || PR [∵ Given]

$\dfrac{OA}{AP} = \dfrac{OC}{CR}$ …………. (2)

From equations (1) and (2), we get :

$\dfrac{OB}{BQ} = \dfrac{OC}{CR}$

In Δ OQR,

$\Rightarrow \dfrac{OB}{BQ} = \dfrac{OC}{CR}$.

We know that,

If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.

∴ BC || QR.

Hence, proved that BC || QR.