Question

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

Answer 1

Let the lines be l₁ and l_2.

Assume that O touches l₁ and l₂ at M and N,

We get,

OM = ON  ......(Radius of the circle)

Therefore,

From the centre ”O” of the circle, it has equal distance from l₁ and l₂.

In Δ OPM and OPN,

OM = ON  ......(Radius of the circle)

∠OMP = ∠ONP  ......(As, Radius is perpendicular to its tangent)

OP = OP .....(Common sides)

Therefore,

ΔOPM = ΔOPN ......(SSS congruence rule)

By C.P.C.T,

∠MPO = ∠NPO

So, l bisects ∠MPN.

Therefore, O lies on the bisector of the angle between l₁ and l₂ .

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