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Prove that the centre of a circle touching two intersecting lines lies on the angle bisector of the lines.
Given:
The centre of a circle touching two intersecting lines.
To do:
We have to prove that the centre of a circle touching two intersecting lines lies on the angle bisector of the lines.
Solution:
Let two tangents $PQ$ and $PR$ are drawn from an external point $P$ to a circle with centre $O$.
Join $OR$ and $OQ$.
In $\triangle POR$ and $\triangle POQ$,
$\angle PRO = \angle PQO = 90^o$ (Tangent at any point of a circle is perpendicular to the radius through the point of contact)
$OR = OQ$ (Radii of the circle)
$OP = OP$ (Common side)
Therefore, by RHS congruency,
$\triangle PRO\ \cong\ \triangle PQO$
This implies,
$\angle RPO = \angle QPO$ ( CPCT)
Hence, $O$ lies on the angle bisector of $PR$ and $PQ$.
Hence proved.