Does the common chord of two circles pass through this point

Question

Well, the question is : Does the common chord of two circles pass through the mid-point of the direct common tangent ?

my approach : I tried to us power of a point with respect to a circle, some geometry and some other random attempts but could not get a good breakthrough at this question .

So, is the given statement true, kindly provide a counter example if its not true.

All help is greatly appreciated.

Answer 1

$$PQ^2=PA\cdot PB = PS^2\to PQ=PS$$

Answer 2

The line you are talking about has also got other mathematical significance and a beautiful name too. For two circles $S_1=0$ and $S_2=0$ where the coefficients of $x^2$ are same in both the equations, the equation of this line is given by $S_1 - S_2 = 0$.

For more, see here

Answer 3

By choosing the radical axis $AB$ for a point on $ R,$ so that $ RQ'=RS;$ even when $Q'RS'$ is not a straight line, even when circles are non-intersecting.