Prove that the centre of a circle touching two intersecting lines lies on the angles biscetor of the lines.

Let l1 and l2 be two intersecting lines. Suppose a circle with centre O touches the lines l1 and l2 at M and N respectively.

∴ OM = ON (Radius of the same circle)

⇒ O is equidistant forml1 and l2

In ΔOPM and ΔOPN,

∠OMP = ∠ONP (Radius is perpendicular to the tangent at the point of contact)

OP = OP (Common)

OM = ON (Radius of the same circle)

∴ ΔOPM ΔOPN (RHS congruence criterion)

⇒ ∠MPO = ∠NPO (CPCT)

⇒ l bisects ∠MPN.

⇒ O lies on the bisector of the angles between l1 and l2i.e., O lies on l.

Thus, the centre of the circle touching two intersecting lines lies on the angle bisector of the two lines.

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