We take as a fact that a circle has 360 degrees, the definition of degree.

Let us take a arbitary triangle as above.
We draw the incircle and the incenter, conected to the vertices:

We now have three triangles:


The sum of three triangles, with each triangle as x is 3x.
Now notice the big triangle is the the three little triangles minus the inner circle, or 360.

Hah... don't you mean "Proof that a triangle has 180 degrees"?
I have a different way but seeing as how I am retarded at latex/asymtote..

W/e changing

Proof that a triangle has exterior angles adding up to 360

let x y and z be the interior angles. By dojo, x+y+z=180

the sum of the exterior angles is 180-x+180-y+180-z or 540-(x+y+z). Plug in x+y+z=180 for 360

Proof of the Pythagorean Theorem.

It's so much easier with a diagram, but here goes:

Draw 4 identical right triangles, each rotated 90 degrees more than the previous. Form a square with them, with a hole in the middle and the hypotenuse as the side. So the area is c^2, because that's the hypotenuse.

We can calculate this another way: the four triangles have area 1/2ab, and the hole in the middle has area (b-a)^2. So: 2ab+b^2+a^2-2ab=a^2+b^2.

Therefore, a^2+b^2=c^2.

THere are hunderds more proofs of the pythagorean theorem! How many can you find?

Hmmm....
Maybe there should be a new thread on pythagoreas

Umm... You can use the alternate interior angles theorem and the corresponding angles theorem to prove that a triangle has 180 degrees.