How many right triangles are there with these conditions?

Question

How many right triangles are there with the following conditions:

the sides $a$, $b$, and $c$ have an integer length (Pythagorean triplets)

the amounts of area and perimeter are the same for each triangle.

Gareth McCaughan · Accepted Answer

Here's another approach which I mention not because it's necessarily better than hexomino's, but because the technique is useful to know.
Every integer right-angled triangle

So

hexomino · Answer

I think there are

Which are the following

Proof

Culver Kwan · Answer

I have a solution, as follows:

Edit: Sorry that this is similar to @hexomino's solution.
2nd edit: How could I forget to spoilorise it?

Paul Panzer · Answer

Here is an economical solution:

How many right triangles are there with these conditions?

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