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Prove that the centre of a 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 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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