RE: Is this a proof of Pythagorean Theorem?

Hi,

Is this a legitimate proof of the Pythagorean theorem?

We know that Pythagorean triples **a**, **b**, **c**, such that a^2 +b^2 = c^2.

We know that that a = 2n+1, b= 2n(n+1), and c = 2n(n+1) + 1 for finding Pythagorean triples.

If you substitute these into the formula a^2 + b^2 = c^2, expand and collect like terms to show they are equal, would this be considered a proof?

Thanks!

Re: Is this a proof of Pythagorean Theorem?

no, The Pythagorean thrm states the **connection** between the side lengths of right triangles and the **algebraic equation** $\displaystyle a^2 + b^2 = c^2$, namely that every right triangle has side lengths satisfying the equation. What you are doing is merely proving that a = 2n+1, b= 2n(n+1), and c = 2n(n+1) + 1 satisfies the **algebraic equation $\displaystyle a^2+b^2=c^2$**. In order to prove the Pythagorean thrm for positive integer triples satisfying the **algebraic equation** $\displaystyle a^2 + b^2 = c^2$, show me that you can use the positive numbers in the tuple (a,b,c) to construct a **right triangle** **OR** show that every right triangle has side lengths satisfying $\displaystyle a^2 + b^2 = c^2$

Re: Is this a proof of Pythagorean Theorem?

Yes, but every side length of a triangle can be constructed using these equations, correct? They would, therefore, be "sides" in general.

Re: Is this a proof of Pythagorean Theorem?

No. Your 'proof' doesn't even mention right angles.

Re: Is this a proof of Pythagorean Theorem?

Quote:

Originally Posted by

**SC313** Yes, but every side length of a triangle can be constructed using these equations, correct? They would, therefore, be "sides" in general.

You need to prove that every side length of a right triangle can be constructed using the equations. If you can do that then your proof would have shown the Pythagorean theorem for **Pythagorean triples only. **. Don't not be confused by the term "Pythagorean triples", they are only positive integer solutions to the equation $\displaystyle a^2+b^2=c^2$. People use the "Pythagorean" prefix to "triples" to imply that these are using these triples in the context of **representing the sides of a right triangle**. If these a,b,c don't represent any object, then they are just triples, numbers, that satisfy $\displaystyle a^2+b^2=c^2$ nothing more.