1. Given a right triangle with sides a, b, and hypotenuse c, the Pythagorean theorem states that a^2 + b^2 = c^2.
2. To prove this theorem, we can create a square with side lengths a + b. The area of this square is (a + b)^2 = a^2 + 2ab + b^2.
3. Next, we can divide this square into two smaller squares, one with side length a and one with side length b. The area of these squares are a^2 and b^2 respectively.
4. Therefore, the area of the two smaller squares combined is a^2 + b^2. This is equal to the area of the larger square, which is a^2 + 2ab + b^2.
5. It follows that a^2 + b^2 = a^2 + 2ab + b^2. By subtracting a^2 from both sides, we get b^2 = 2ab. Similarly, by subtracting b^2 from both sides, we get a^2 = 2ab.
6. Dividing both sides by 2, we have a^2/2 = ab/2 and b^2/2 = ab/2. This simplifies to a^2/2 = ab and b^2/2 = ab.
7. Since a^2/2 = ab and b^2/2 = ab, we can add these equations together to get a^2/2 + b^2/2 = 2ab.
8. Simplifying, we have (a^2 + b^2)/2 = 2ab. Multiplying both sides by 2, we get a^2 + b^2 = 2ab.
9. Therefore, the Pythagorean theorem is proven: a^2 + b^2 = c^2 where c is the hypotenuse of the right triangle.
pythagorean theorem proofs answers connxus
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