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Question

Question 3
Prove that the centre of circle touching two intersecting lines lies on the angle bisector of the lines.

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Solution

Given two tangents PQ and PR are drawn from external point P to a circle with centre O.

To prove: Centre of a circle touching two intersecting lines lies on the angle bisector of the lines.
Construction Join OR and OQ.
In POR and POQ
PRO = PQO = 90
[ tangents at any point of a circle is perpendicular to the radius through the point of contact]
OR =OQ [Radii of some circle]
Since, OP is common.
ΔPROΔPQO [RHS]
Hence RPO = QPO [by CPCT]
Thus, O lies on angle bisecter of PR and PQ.
Hence proved.

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