# Thread: Proving . . .

1. ## Proving . . .

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.

2. ## Trying to help but...

Hi!
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?

3. See this (I know it is not accurate but it helps)

4. If you know it, written form will be sufficient for me.

5. 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.

6. ## Well, does this make sence

Is this somewhat understandable?

7. Simplest is the following...

Triangle TBE is isosceles,
from this you can show |BM|=|MA|

Again, CD needs to be a centreline of the smaller circle.

8. A general proof, when CD is not a diameter of the smaller circle
can be derived from the attached sketch.

9. Here is a geometric proof.