Proving . . .

[x-mather]

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.

 

[chrisr]

[attachment=0:z5rbm0e3]intersection=M.jpg[/attachment:z5rbm0e3]

The large circle is "I', the small one is "k".
If you draw a second large circle "I", you see the symmetry of the geometry.

Can you complete the analysis with this?

 

[x-mather]

Ok, but it is possible to solve it in a written form. Thank you.

 

[Denis]

Ok, but it is possible to solve it in a written form.

Well, go ahead and do it then; show us the results...

 

[x-mather]

I understand drawing perfectly but I don't know to prove it in a written form. I am so stupid. Please write me the proof. I have spended 2 hours by thinking and trying to write it. Please you are my last chance.

 

[chrisr]

We can only do this step by step,
as writing a full proof will be useless to you unless you practice
until you've mastered it.
Hence, you must submit work to the thread after each step.

 

[chrisr]

Your first step is to draw a line which will contain the centre of both circles.
Pick 2 points to be the circle centres.
Using a compass, draw a small circle (not very small of course) similar to the blue one in the attachment.
Then draw the vertical diameter of this circle.
Let us know when you have this done.

 

[x-mather]

I have done it but I do not understand how will this help me?

 

[chrisr]

[attachment=0:u8a22y19]circles_a.jpg[/attachment:u8a22y19]

Sketch the above with your compass.
Writing a purely algebraic answer is pointless to me.

 

[chrisr]

[attachment=0:balzfnup]circles_b.jpg[/attachment:balzfnup]

Then draw the red circle (identical to the small one) on the centreline of the large one as shown.
Draw the vertical green line.

 

[chrisr]

[attachment=0:xvj3bpq3]circles_c.jpg[/attachment:xvj3bpq3]

Add the second large red circle and draw the new blue line shown.

 

[chrisr]

If the geometry isn't clear, don't even think about writing it in code.
You would learn nothing.
Your question does not ask for a non-geometric proof.
You only need a compass, pencil, coloured pens and ruler for this.

Your move.

 

[x-mather]

OK thank you.

 

[chrisr]

Are you sure you can write a proof now?

It's this type of diagram from which you "word" the proof.

There are other ways to derive a proof from the initial diagram of course.

I doubt you were expected to write an "algebraic" proof without the visual construction.
This is a geometric problem, for which a written proof is finally derived from the constructions illustrated.

Try to submit a written proof and we will add to it if necessary.