<s>[INST] Do four natural numbers exist which satisfy these constraints? Do four natural numbers $a,b,c$ and $d$ exist such that the following three conditions are true?
$$a^2+b^2+2d^2=c^2$$
$$\sqrt{a^2+d^2}\in\mathbb{N}$$
$$\sqrt{b^2+d^2}\in \mathbb{N}$$
 [/INST] For any Pythagorean triple $(x,y,z)$ (such  that $x^2+y^2=z^2$)
denote
$$
a=x^2,\\
b=y^2,\\
c=z^2,\\
d=xy.
$$
Then
$$
a^2+b^2+2d^2 = a^2+b^2+2ab = (a+b)^2=z^4=c^2,
$$
$$
a^2+d^2=x^4+x^2y^2=x^2(x^2+y^2)=(xz)^2,
$$
$$
b^2+d^2=y^4+x^2y^2=y^2(y^2+x^2)=(yz)^2.
$$
</s>