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In a PQR, ...

Question

In a $△ P Q R$ , $L$ and $M$ are two points on the base $Q R$ such that $∠ L P Q = ∠ Q R P$ and $∠ R P M = ∠ R Q P$ .Prove that $Q L \times R M = P L \times P M$

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Solution

In $△ P Q L$ and $△ R M P$
$∠ L P Q = ∠ Q R P$ (given)
$∠ R Q P = ∠ R P M$ (given)
$\Rightarrow △ P Q L \sim △ R M P$ (AA similarity)
$\Rightarrow \frac{P Q}{R P} = \frac{Q L}{P M} = \frac{P L}{R M}$
$\Rightarrow Q L \times R M = P L \times P M$
Hence proved.

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