My proof involving Pythagorean’s Theorem

[Jadehaan]

I REALIZE THAT THIS IS A DUPLICATE. MY APOLOGIZES. PLEASE IGNORE.

Homework Statement

Let a, b, and c be lengths of sides of triangle T, where a ≤ b ≤ c.
Prove that if T is a right triangle, then (abc)²=(c⁶-a⁶-b⁶)/3

Homework Equations

If T is a right triangle, then Pythagorean’s Theorem states:
The sum of the squares of the lengths of the sides of a right triangle is equal to the square of the length of the hypotenuse. That is a²+b²=c², where c is the hypotenuse.

The Attempt at a Solution

We assume the given equation and using Pythagorean’s Theorem, we obtain solutions for c² and c⁶:
We substitute these results into the original equation.
This produces an equation where the left hand side is identical to the right hand side.
Since these terms are equal, it follows that the original equation holds true for a right triangle.

This is what I have. I am curious to if the proof is correct/acceptable.

Thanks for any feedback.

 

[mighty2000]

Thank you for your post. Your attempt at a solution is on the right track, but there are a few errors and missing steps in your proof.

Firstly, in your statement of the Pythagorean's Theorem, it should be a^2 + b^2 = c^2, not a2+b2=c2. This is important to note because it affects the rest of your proof.

Next, when you substitute c^2 and c^6 into the original equation, you should end up with (abc)^2 = (c^6-a^6-b^6)/3, not (abc)^2 = (c^6-a^6-b^6). The 3 in the denominator is important and should not be left out.

Additionally, you should explain how you obtained the solutions for c^2 and c^6 using Pythagorean's Theorem. This can be done by showing the steps of the theorem and substituting a^2 + b^2 for c^2.

Lastly, you need to show how the left hand side is identical to the right hand side, as you have mentioned. This can be done by showing the steps of simplifying the equation and arriving at the same result on both sides.

Overall, your proof is on the right track but needs some clarification and additional steps. I hope this helps. Keep up the good work!