Question

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

Answer 1

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.