Proof That Common Tangent From Meeting Point of Two Circles Bisects Other Common Tangent
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.