In the figure,$A, B$ and $C$ are points on $OP, OQ$ and $OR$ respectively such that $AB \parallel PQ$ and $AC \parallel PR$. Show that $BC \parallel QR$.
(N/A) In $\Delta POQ$,$AB \parallel PQ$.
Therefore,$\frac{OA}{AP} = \frac{OB}{BQ}$ (Basic Proportionality Theorem) $...(i)$
In $\Delta POR$,$AC \parallel PR$.
Therefore,$\frac{OA}{AP} = \frac{OC}{CR}$ (Basic Proportionality Theorem) $...(ii)$
From $(i)$ and $(ii)$,we obtain:
$\frac{OB}{BQ} = \frac{OC}{CR}$
Therefore,$BC \parallel QR$ (By the converse of the Basic Proportionality Theorem).
Therefore,$\frac{OA}{AP} = \frac{OB}{BQ}$ (Basic Proportionality Theorem) $...(i)$
In $\Delta POR$,$AC \parallel PR$.
Therefore,$\frac{OA}{AP} = \frac{OC}{CR}$ (Basic Proportionality Theorem) $...(ii)$
From $(i)$ and $(ii)$,we obtain:
$\frac{OB}{BQ} = \frac{OC}{CR}$
Therefore,$BC \parallel QR$ (By the converse of the Basic Proportionality Theorem).
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