## Make {x^2 + y^2 ± xy, x^2 + z^2 ± xy, y^2 + z^2 ± xy} squares

(1)

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

Here are some solutions:

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

$\{ \, 22, \; 70, \; 41, \; 62, \; 25, \; 71 \, \}$
$\{ \, 468, \; 660, \; 316, \; 588, \; 100, \; 476 \, \}$
$\{ \, 2444, \; 2924, \; 1121, \; 2716, \; 289, \; 1631 \, \}$
$\{ \, 8140, \; 9100, \; 2876, \; 8660, \; 676, \; 4124 \, \}$
$\{ \, 21186, \; 22866, \; 6121, \; 22074, \; 1369, \; 8711 \, \}$
$\{ \, 47012, \; 49700, \; 11516, \; 48412, \; 2500, \; 16316 \, \}$
$\{ \, 93208, \; 97240, \; 19841, \; 95288, \; 4225, \; 28031 \, \}$
$\{ \, 169884, \; 175644, \; 31996, \; 172836, \; 6724, \; 45116 \, \}$
$\{ \, 290030, \; 297950, \; 49001, \; 294070, \; 10201, \; 68999 \, \}$

Find other solutions where   $x < 290030$

(2)

Find positive integers   $(x, y, z)$   such that

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