## Make {x^2 + 2xy + y^2, x^2 + 3xz + z^2, y^2 – 5yz + z^2} squares … Part 1

$x^2 \; + \; 2 \, x \, y \; + \; y^2 \; = \; A^2$
$x^2 \; + \; 3 \, x \, z \; + \; z^2 \; = \; B^2$
$y^2 \; - \; 5 \, y \, z \; + \; z^2 \; = \; C^2$

where   $x, y, z$   are positive integers.

Here are the first few solutions:

$\{x, \; y, \; z, \; A, \; B, \; C \}$

$\{112, \; 231, \; 48, \; 343, \; 176, \; 15 \}$
$\{240, \; 1335, \; 176, \; 1575, \; 464, \; 799 \}$
$\{264, \; 744, \; 143, \; 1008, \; 451, \; 205 \}$
$\{456, \; 552, \; 95, \; 1008, \; 589, \; 227 \}$

Find the next few solutions.