There are many different tasks in pictures on the Internet. I found one picture and it interested me.
And two questions. The first. Did I write this system correctly? That is, to reformulate the problem in the form of two right triangles.
$$ \left\{\!\begin{aligned} & X^2+Y^2=Z^2 \\ & A^2+B^2=C^2 \\ & 2XY=AB \\ & 2Y+2Z=A+B+C \end{aligned}\right. $$ And the second question... these are some ideas for solving such systems? Not numerical, but rather parametrization of solutions.
Where did you find this meme/problem?
The problem is asking for a right triangle and an isoceles triangle with the same area and permimeter. This was asked and solved only recently by Hirakawa and Matsumura, see https://arxiv.org/abs/1809.09936. They showed using quite advanced techniques that there is a unique solution. They give the equations they use there, which are different to yours but could be equivalent.
See also the video abstract: https://www.youtube.com/watch?v=iln_m7zymtw