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.