Two tangents $TP$ and $TQ$ are drawn to a circle with centre $O$ from an external point $T$. Prove that $\angle PTQ = 2 \angle OPQ$.

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Question

Two tangents $TP$ and $TQ$ are drawn to a circle with centre $O$ from an external point $T$. Prove that $\angle PTQ = 2 \angle OPQ$.

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Accepted Answer

Step 1: $TP = TQ$
$\Rightarrow \angle TPQ = \angle TQP$
Step 2: Let $\angle PTQ$ be $	heta$
$\Rightarrow \angle TPQ = \angle TQP = \frac{180° - 	heta}{2} = 90° - \frac{	heta}{2}$
Now $\angle OPT = 90°$
Step 3: $\Rightarrow \angle OPQ = 90° - (90° - \frac{	heta}{2}) = \frac{	heta}{2}$
$\angle PTQ = 2 \angle OPQ$

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