1. ## Circle Theorem

Hi I have attached the pdf file which I have a question about. On the first page, where it says:

Observe that the proof is similar if the center O is not inside triangle ABC. Since....

Why is the exterior side of angle COB = 2A and not the interior angle? The circle theorem gives the angle formed in the triangle doesn't it?

2. If you just want to prove that exterior angle COB = 2 x angle A,
consider a line AO and extend it to D.
AOD divides angle A into A1 (CAO) and A2 (BAO).
by theorem of exterior angles of triangles, you see that
angles COD = 2x A1 and angles BOD = 2 x A2
hence exterior angle COB = 2 x (A1 + A2) = 2A

3. Originally Posted by ukorov
If you just want to prove that exterior angle COB = 2 x angle A,
consider a line AO and extend it to D.
AOD divides angle A into A1 (CAO) and A2 (BAO).
by theorem of exterior angles of triangles, you see that
angles COD = 2x A1 and angles BOD = 2 x A2
hence exterior angle COB = 2 x (A1 + A2) = 2A
Thanks.

What I wanted to know was that, by the circle theorem, the interior angle COB should be 2A, the one forming the triangle. Not the one outside that they have taken.

Why is this? The one outside is not forming a triangle.

4. Originally Posted by Aquafina
Thanks.

What I wanted to know was that, by the circle theorem, the interior angle COB should be 2A, the one forming the triangle. Not the one outside that they have taken.

Why is this? The one outside is not forming a triangle.
you are happy with the fact that exterior angle COB = 2 x angle CAB, right?

therefore interior angle COB = 360 - 2A, and hence both angles COD and BOD = 180 - A

and then sin (180 - A) = sin A. note that here it does not mean that 180 - A = A. It only means that the Sine ratio of both are equal.

5. Originally Posted by ukorov
you are happy with the fact that exterior angle COB = 2 x angle CAB, right?

therefore interior angle COB = 360 - 2A, and hence both angles COD and BOD = 180 - A

and then sin (180 - A) = sin A. note that here it does not mean that 180 - A = A. It only means that the Sine ratio of both are equal.
Hi, no what I am not happy with is the exterior angle being 2A, I thought it should be the interior angle, according to the circle theorem