## Make {(x^2 + 3xy + y^2), (x^2 + 4xz + z^2), (y^2 + 5yz + z^2)} squares

Find positive integers   $x, \; y, \; z$   such that

$x^2 \; + \; 3 \, x \, y \; + \; y^2$
$x^2 \; + \; 4 \, x \, z \; + \; z^2$
$y^2 \; + \; 5 \, y \, z \; + \; z^2$

Here’s one solution:

$(x, \; y, \; z) \; = \; (32, \; 21, \; 40)$

$32^2 \; + \; 3 \,(32) \,(21) \; + \; 21^2 \; = \; 59^2$
$32^2 \; + \; 4 \,(32) \,(40) \; + \; 40^2 \; = \; 88^2$
$21^2 \; + \; 5 \,(21) \,(40) \; + \; 40^2 \; = \; 79^2$

Find other solutions.