In the given figure, MN is the common chord of two intersecting circles and AB is their common tangent.
Prove that the line NM produced bisects AB at P.
From P, AP is the tangent and PMN is the secant for first circle.
∴ `AP^2 = PM × PN` …… (i)
Again from P, PB is the tangent and PMN is the secant for second circle.
∴ `PB^2 = PM × PN` ……..(ii)
From (i) and (ii)
`AP^2 = PB^2`
⇒ AP = PB
Therefore, P is the midpoint of AB.