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

Find three positive integers   $x, y, z$   such that   $(x, y, z) = 1$   and

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

Here are the first few solutions:

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

$\{ 112, \; 259, \; 48, \; 147, \; 176, \; 85 \}$
$\{ 456, \; 888, \; 95, \; 432, \; 589, \; 613 \}$
$\{ 1488, \; 3216, \; 527, \; 1728, \; 2201, \; 1465 \}$
$\{ 4720, \; 9520, \; 1239, \; 4800, \; 6431, \; 5761 \}$
$\{ 11088, \; 22995, \; 3344, \; 11907, \; 15664, \; 12469 \}$
$\{ 23352, \; 44520, \; 4031, \; 21168, \; 29051, \; 33181 \}$
$\{ 30096, \; 59499, \; 6992, \; 29403, \; 39824, \; 38845 \}$
$\{ 42784, \; 80416, \; 6303, \; 37632, \; 51761, \; 63025 \}$
$\{ 66352, \; 127699, \; 12528, \; 61347, \; 83984, \; 92005 \}$
$\{ 127920, \; 241995, \; 20336, \; 114075, \; 156784, \; 185389 \}$

Find the next few solutions.