## Make {x^2 + 3xy + y^2, x^2 + 5xz + z^2, y^2 + 7yz + z^2} squares

#1

$x^2 \; + \; 3 \,x \,y \; + \; y^2 \; = \; A^2$
$x^2 \; + \; 5 \,x \,z \; + \; z^2 \; = \; B^2$
$y^2 \; + \; 7 \,y \,z \; + \; z^2 \; = \; C^2$

Here are the first few solutions:

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

$\{ \, 104, \; 525, \; 120, \; 671, \; 296, \; 855 \, \}$
$\{ \, 384, \; 493, \; 528, \; 979, \; 1200, \; 1531 \, \}$
$\{ \, 1736, \; 2397, \; 17112, \; 4609, \; 21080, \; 24201 \, \}$

#2

$x^2 \; + \; 3 \,x \,y \; + \; y^2 \; = \; A^2$
$x^2 \; + \; 5 \,x \,z \; + \; z^2 \; = \; B^2$
$y^2 \; - \; 7 \,y \,z \; + \; z^2 \; = \; C^2$

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

$\{ \, 3, \; 7, \; 1, \; 11, \; 5, \; 1 \, \}$
$\{ \, 15, \; 259, \; 13, \; 281, \; 37, 209 \, \}$
$\{ \, 192, \; 1001, \; 120, \; 1271, \; 408, \; 419 \, \}$
$\{ \, 315, \; 1079, \; 153, \; 1511, \; 603, \; 179 \, \}$
$\{ \, 1755, \; 2831, \; 297, \; 5099, \; 2403, \; 1489 \, \}$

#3

$x^2 \; - \; 3 \,x \,y \; + \; y^2 \; = \; A^2$
$x^2 \; - \; 5 \,x \,z \; + \; z^2 \; = \; B^2$
$y^2 \; + \; 7 \,y \,z \; + \; z^2 \; = \; C^2$

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

the first few solutions:

$\{ \, 8, \; 21, \; 40, \; 1, \; 8, \; 89 \, \}$
$\{ \, 312, \; 817, \; 1664, \; 11, \; 520, \; 3599 \, \}$
$\{ \, 1368, \; 3901, \; 12920, \; 1039, \; 8968, \; 23129 \, \}$

#4

$x^2 \; + \; 3 \,x \,y \; + \; y^2 \; = \; A^2$
$x^2 \; - \; 5 \,x \,z \; + \; z^2 \; = \; B^2$
$y^2 \; + \; 7 \,y \,z \; + \; z^2 \; = \; C^2$

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

$\{ \, 143, \; 27, \; 819, \; 181, \; 325, \; 909 \, \}$
$\{ \, 285, \; 80, \; 1729, \; 395, \; 779, \; 1991 \, \}$
$\{ \, 481, \; 1032, \; 5957, \; 1669, \; 4625, \; 8921 \, \}$
$\{ \, 989, \; 960, \; 8385, \; 2179, \; 5461, \; 11295 \, \}$

#5

$x^2 \; - \; 3 \,x \,y \; + \; y^2 \; = \; A^2$
$x^2 \; - \; 5 \,x \,z \; + \; z^2 \; = \; B^2$
$y^2 \; - \; 7 \,y \,z \; + \; z^2 \; = \; C^2$

the first few solutions:

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

$\{ \, 5, \; 15, \; 1, \; 5, \; 1, \; 11 \, \}$
$\{ \, 231, \; 680, \; 35, \; 211, \; 119, \; 545 \, \}$
$\{ \, 741, \; 2135, \; 65, \; 601, \; 559, \; 1895 \, \}$
$\{ \, 819, \; 2432, \; 143, \; 781, \; 325, \; 1871 \, \}$
$\{ \, 896, \; 2613, \; 112, \; 779, \; 560, \; 2189 \, \}$
$\{ \, 1615, \; 4680, \; 171, \; 1355, \; 1121, \; 4041 \, \}$

Find the next few solutions