Some results of Apollonius - intersecting circles
The point P cuts L into two half-lines. If N cuts both, then P lies inside N, so that N also cuts
both parts of M. Otherwise, N cuts one of the half-lines of each of L
and M.
Ypu can vary the circle N by moving its centre C, or the point D.
You can vary the line M by moving the point R.
You can vary the circles tangent to L and M by moving A or B.
intersecting circles