Ideas in Geometry/Instructive examples/Proving Pythagorean Theorem Using Similar Angels
On the picture of the triangle with the segment of ab cut in the middle of the triangle, we see that the small triangle with sides bc, b^2 and ab were multiplied by b. If we divide this triangle by b again, the sides would become side c, side b and side a. The other small triangle that is attached was multiplied by a, forming sides ab, ac, and a^2. Again if we divide these triangles up by a, the sides would be side b, side c and side a. The angle across from Side length of b should both have the same angle. Same goes for the angle across from side length a, and c. For example, the angles across from sides b in both the small triangles should have the same angle. The angle across from sides a in both the triangles should also have the same angles. Same goes for angles across from side c.
Since the angles across from sides a, and the angles across from sides b are the same, the angles across from side C should be 90 degrees. This means that there are angles of 30, 60 and 90 degrees in each of the small triangles, which proves that this that the segment ab splitting the whole triangle is actually cutting a 90 degrees angle. This proves that the triangle is 90 degrees. Then we can assume that the sides of the whole triangle are equal to the triangle of the big triangle. There for the hypotenuse of the triangle with the segment ab, should be a^2 + b^2, which should be equal to the hypotenuse of the other large triangle which is c^2. So a^2+b^2=c^2