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Question
Prove that the centre of a circle touching two intersecting lines lies on the angle bisector of the lines.
Solution
Given: Two tangents PQ and PR are drawn from an external points P to a circle with centre O.
To prove: Centre of a circle touching two intersecting lines lies on the angle bisector of angle formed by tangents.
Construction: Join OR and OQ.
In ∆POR and ∆POQ,
∠PRO = ∠PQO = 90° ...[Tangent at any point of a circle is perpendicular to the radius through the point of contact]
OR = OQ ...[Radii of same circle]
Since, OP is common
∴ ∆PRO ≅ ∆PQO ...[RHS]
Hence, ∠RPO = ∠QPO ...[By CPCT]
Thus, O lies on angle bisector of PR and PQ.
Hence proved.
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