Question

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

Answer 1

Assume that O touches l₁ and l₂ at M and N , then we get as ,

⇒ OM = ON ( As it is the radius of the circle )

- Therefore ,From the centre ''O'' of the circle , it has equal distance from l₁ & l₂.

Now , In Δ OPM & OPN ,

⇒ OM = ON ( Radius of the circle )

⇒Angle - OMP = Angle -ONP ( As , Radius is perpendicular to its tangent )

⇒ OP = OP ( Common sides )

Therefore , Δ OPM = ΔOPN ( S S S congruence rule )

By C.P.C.T ,

⇒ Angle MPO = Angle NPO

So , l bisects angle MPN.

Threfore , O lies on the bisector of the angle between l₁ & l₂ .

Hence , we prove that ⇒ the centre of a circle touching two intersecting lines lies on the angle bisector of the lines.