The diagram below shows common tangents PQ and Abto two circles.

The tangent PQ bisects the tangent AB.

Two see this draw a line between the centre of each circle. This line must cut the line PQ at right angles. Draw two other lines from the centre of each circle to the points A and B as shown

AQ=QS since they are tangents to the circle on the left from a common point, and QB=QS similarly for the circle on the right. Hence AQ=QB.