⇒ A circle wit centre '' O '''.
⇒ And , also O touches Line - l₁ and l₂ .
⇒ l₁ and l₂ lines intersect at each other
To Prove :-
The centre of a circle touching two intersecting lines lies on the angle bisector of the lines.
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.