We prove that we can construct using Pythagorean triples. Take any Pythagorean triple $(x, y, z)$ (not necessarily primitive). Let
$$a=x\left|4 y^{2}-z^{2}\right|, b=y\left|4 x^{2}-z^{2}\right|, c=4 x y z,$$

then we have
$$\begin{aligned} a^{2}+b^{2} & =x^{2}\left(3 y^{2}-x^{2}\right)^{2}+y^{2}\left(3 x^{2}-y^{2}\right)^{2} \\ & =x^{6}+3 x^{2} y^{4}+3 x^{4} y^{2}+y^{6} \\ & =\left(x^{2}+y^{2}\right)^{3}=\left(z^{3}\right)^{2} \\ a^{2}+c^{2} & =x^{2}\left(4 y^{2}+z^{2}\right)^{2} \\ b^{2}+c^{2} & =y^{2}\left(4 x^{2}+z^{2}\right)^{2} \end{aligned}$$

Since there are infinitely many Pythagorean triples, there are infinitely many such triples $(a, b, c)$.
In particular, when $x=3, y=4, z=5$, we get $a=117, b=44, c=240$, and
$$117^{2}+44^{2}=125^{2}, 117^{2}+240^{2}=267^{2}, 44^{2}+240^{2}=244^{2} .$$