Question

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

Answer 1

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 $\angle$POR and $\angle$POQ
$\angle$ PRO = $\angle$ PQO = ${90}^{\circ }$
[ 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.
$\therefore \Delta PRO\cong \Delta PQO$ [RHS]
Hence $\angle$ RPO = $\angle$ QPO [by CPCT]
Thus, O lies on angle bisecter of PR and PQ.
Hence proved.