Proof of the Pythagorean Theorem
The blue, purple, green, and pink triangles are all identical triangles with two legs of length a and b and a hypotenuse of length c. As shown on the left, the four triangles can combine to form a square with length c (the hypotenuse) on each side. Therefore, the area of square (also known as c^2) is given in the middle. The group of triangles on the right are color coded to show that they are the same triangles as on the left, just simply rearranged to show that they can form two smaller squares, one with sides of length a and the other with sides of length b. Therefore, the squares formed on the right are of areas equal to a^2 and b^2, respectively. The Pythagorean Theorem is proved by noting that the area of c^2 is equal to the sum of the areas a^2 and b^2. You can manipulate the lengths of sides a and b by using the slider bars located in quadrant II.
Credit for demonstration of this theorem and how to create objects in GeoGebra goes to Doug Stevens (https://www.youtube.com/watch?v=I34-TRgBuN8&noredirect=1).