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

"

Given:

$A, B$ and $C$ are points on $OP, OQ$ and $OR$ respectively such that $AB \| PQ$ and $AC \| PR$.

To do:

We have to show that $BC \| QR$.

Solution:

We know that,

If a line divides two sides of a triangle proportionally, then it is parallel to the third side.

In $\triangle POQ, AB \| PQ$,

This implies,

$\frac{OP}{PA}=\frac{OQ}{QB}$.........(i)

In $\triangle POR, AC \| PR$,

This implies,

$\frac{OP}{PA}=\frac{OR}{RC}$.........(ii)

From (i) and (ii), we get,

$\frac{OQ}{QB}=\frac{OR}{RC}$

By converse of B.P.T.,

$BC \| QR$

Hence proved.

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