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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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