Reading the problem over and over and canīt quite grash it. C;D are antipods of large circle? And inside that circle are two other circles?
Circle k(S; r)is touching point A of line AB. Circle l(T; s)is touching point B of line AB and intersects circle k in the edge points C, D of its diameter. Prove that the intersection M of lines CD and AB is the centre of line AB.
Here is a sketch of the geometry.
The large blue circle has the same radius as circle k.
Using the radius of the smaller circle,
next draw right-angled triangles to show |TF|=|TG|.
The show |BM|=|BA|
I had to shrink my sketch and it has become a little skewed.
Those lines are meant to be perpendicular.
This only works if CD is a centreline.