Can I prove Pythagoras therorem by concatenating four right triangles? I was considering ways to visually prove Pythagoras' theorem.
I was reasoning if I could concatenate the same right triangle four times, as if it was reflected from both axis.

Why can't I just calculate the area of this diamond to know c?
What am I doing wrong? Why does it make no sense?
Please, excuse me if I'm doing an obvious mistake or wrong assumption.
Thanks.
 A: Problem
The shape you have drawn is a rhombus. The area of a rhombus is defined to be $\frac{pq}2$ where $p$ and $q$ are the lengths of both the diagonals.
Therefore, the area of the shape you have drawn is calculated by considering the fact that $p=2b$ and $q=2a$. And so the area would be $\frac{(2a)(2b)}2=2ab$.
We can of course verify this by calculating the area of one of the right angle triangles and then multiplying by 4. The area of one triangle is $\frac{ab}2$ and so multiplied by 4 gives us the same answer for the total area as $2ab$.
However, this doesn’t help us to find $c$, since the formula for the area is independent of $c$
Note: c is not independent of the area. We just need Pythagoras’ Theorem to show that this is the case. Since we are trying to proof Pythagoras’ Theorem, we must use a formula in terms of a and b which make the area independent of c.
So you haven’t written anything explicitly wrong. The only problem is that, this doesn’t prove Pythagoras’ theorem.
Expanding upon something correctly stated in the comments, we also have the problem of the fact that we are stating the formula for the area of a rhombus without any explicit proof. And so this is also a consideration we need to make before even making an attempt to use the formula for a proof.
