Do the centre of a circle touching two intersecting lines lies on the angle bisector of the lines

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

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.

∴ΔPRO≅ΔPQO [RHS]

Hence ∠ RPO = ∠ QPO [by CPCT] Thus, O lies on angle bisecter of PR and PQ. Hence proved.