Two circles touch each other internally. Show that the tangents drawn to the two circles from any point on the common tangent are equal in length.

We know that,

If two tangents are drawn to a circle from an exterior point, the tangents are equal in length.

Let Q be the point on the common tangent from which, two tangents QA and QP are drawn to the circle with centre O.

So, QA = QP ………. (1)

Similarly, from point Q, QB and QP are two tangents to the circle with centre O’

So, QB = QP ………. (2)

From (1) and (2), we have

QA = QB

Hence, proved that the tangents are equal.