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

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Let the lines be land l2.

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 & l₂.

In Δ OPM & OPN,

OM = ON (Radius of the circle)

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

OP = OP (Common sides)

Therefore,

Δ OPM = 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 & l₂ .

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

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