In figure, if ABIIDC and AC, PQ intersect each other at the point O. Prove that OA.CQ = OC.AP.

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Solution

Given AC and PQ intersect each other at the point O and AB II DC.
To prove, OA.CQ = OC. AP
Proof in $Δ A O P$ and $Δ C O Q$ , $∠ A O P = ∠ C O Q$ [vertically opposite angles]
[since, AB II DC and PQ is transversal, so alternate angles]
$∴ Δ A O P \sim Δ C O Q$ [by AAA similarity criterion]
Then, $\frac{O A}{O C} = \frac{A P}{C Q}$ [since, corresponding sides are propotion]
$\Rightarrow$ OA.CQ = OC.AP

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