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

Explanation

Given Two tangents PQ and PR are drawn from an 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 `DeltaPORand DeltaPOQ`
`anglePRO=anglePQO=90^(@)`
[tangent 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.
`:.DeltaPRO~=DeltaPQO` [RHS]
Hence, `angleRPO=angleQPO` [by CPCT]
Thus, O line on angle bisecter of PR and PQ.