Question

Two circles with centres O and O' touch each other at point L. Prove that, a tangent through L bisects the common tangent AB of the two circles at point M.

Given: AB is a common tangent of the two circles that touch each other at point L. ML is a tangent through point L.

To prove: M is a mid-point of the tangent AB or MA = MB.

Proof: From the figure,

M is an external point that draws two tangents, MA and ML to the circle with the centre O.

So, `square` = `square`  ......(i)

Similarly, M draws two tangents ML and MB to the circle with the centre O'.

So, `square` = `square`  ......(ii)

From the equations (i) and (ii),

`square` = `square`

Hence, the tangent at the point L, bisects the common tangent, AB of the two circles at point M.

Answer 1

Given: AB is a common tangent of the two circles that touch each other at point L. ML is a tangent through point L.

To prove: M is a mid-point of the tangent AB or MA = MB.

Proof: From the figure,

M is an external point that draws two tangents, MA and ML to the circle with the centre O.

So, MA = ML  ......(i)

Similarly, M draws two tangents ML and MB to the circle with the centre O'.

So, ML = MB  ......(ii)

From the equations (i) and (ii),

MA = MB

Hence, the tangent at the point L, bisects the common tangent, AB of the two circles at point M.