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Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.
We have to prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.
Let $TS$ be a tangent to the circle with centre $O$ at $P$.
Draw a line $OR$ which intersects the circle at $Q$ and meets the tangent $TS$ at $R$.
$OP = OQ$ (radii of the circle)
$OP$ is less than all those lines which can be drawn from $O$ to $TS$.
$OP$ is the shortest such line.
$OP$ is perpendicular to $TS$.
Perpendicular through $P$ will pass through the centre of the circle.
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