A^2 + B^2 + C^2 = D^2 + E^2

$a^2 \; + \; b^2 \; + \; (a^2+b^2) \; = \; (a+b)^2 \; + \; (a-b)^2$

Also true is,

$(a \, n)^2 + (b \, n)^2 \;+ (a^2+b^2) \; = \; (a + b \, n)^2 \; + \; (a \, n - b)^2$

Let   $a^2 \; + \; b^2 \; = \; c^2$

$(a \, n)^2 \; + \; (b \, n)^2 \; + \; c^2 \; = \; (a + b \, n)^2 \; + \; (a \, n - b)^2$

Let’s take all primitive Pythagorean triples with   $c \; \leq \; 300$,   for   n = 1

$3^2 + 4^2 + 5^2 \; = \; 7^2 + 1^2 \; = \; 60$
$5^2 + 12^2 + 13^2 \; = \; 17^2 + 7^2 \; = \; 338$
$8^2 + 15^2 + 17^2 \; = \; 23^2 + 7^2 \; = \; 578$
$7^2 + 24^2 + 25^2 \; = \; 31^2 + 17^2 \; = \; 1250$
$20^2 + 21^2 + 29^2 \; = \; 41^2 + 1^2 \; = \; 1682$
$12^2 + 35^2 + 37^2 \; = \; 47^2 + 23^2 \; = \; 2738$
$9^2 + 40^2 + 41^2 \; = \; 49^2 + 31^2 \; = \; 3362$
$28^2 + 45^2 + 53^2 \; = \; 73^2 + 17^2 \; = \; 5618$
$11^2 + 60^2 + 61^2 \; = \; 71^2 + 49^2 \; = \; 7442$
$16^2 + 63^2 + 65^2 \; = \; 79^2 + 47^2 \; = \; 8450$
$33^2 + 56^2 + 65^2 \; = \; 89^2 + 23^2 \; = \; 8450$
$48^2 + 55^2 + 73^2 \; = \; 103^2 + 7^2 \; = \; 10658$
$13^2 + 84^2 + 85^2 \; = \; 97^2 + 71^2 \; = \; 14450$
$36^2 + 77^2 + 85^2 \; = \; 113^2 + 41^2 \; = \; 14450$
$39^2 + 80^2 + 89^2 \; = \; 119^2 + 41^2 \; = \; 15842$
$65^2 + 72^2 + 97^2 \; = \; 137^2 + 7^2 \; = \; 18818$
$20^2 + 99^2 + 101^2 \; = \; 119^2 + 79^2 \; = \; 20402$
$60^2 + 91^2 + 109^2 \; = \; 151^2 + 31^2 \; = \; 23762$
$15^2 + 112^2 + 113^2 \; = \; 127^2 + 97^2 \; = \; 25538$
$44^2 + 117^2 + 125^2 \; = \; 161^2 + 73^2 \; = \; 31250$
$88^2 + 105^2 + 137^2 \; = \; 193^2 + 17^2 \; = \; 37538$
$17^2 + 144^2 + 145^2 \; = \; 161^2 + 127^2 \; = \; 42050$
$24^2 + 143^2 + 145^2 \; = \; 167^2 + 119^2 \; = \; 42050$
$51^2 + 140^2 + 149^2 \; = \; 191^2 + 89^2 \; = \; 44402$
$85^2 + 132^2 + 157^2 \; = \; 217^2 + 47^2 \; = \; 49298$
$119^2 + 120^2 + 169^2 \; = \; 239^2 + 1^2 \; = \; 57122$
$52^2 + 165^2 + 173^2 \; = \; 217^2 + 113^2 \; = \; 59858$
$19^2 + 180^2 + 181^2 \; = \; 199^2 + 161^2 \; = \; 65522$
$57^2 + 176^2 + 185^2 \; = \; 233^2 + 119^2 \; = \; 68450$
$104^2 + 153^2 + 185^2 \; = \; 257^2 + 49^2 \; = \; 68450$
$95^2 + 168^2 + 193^2 \; = \; 263^2 + 73^2 \; = \; 74498$
$28^2 + 195^2 + 197^2 \; = \; 223^2 + 167^2 \; = \; 77618$
$84^2 + 187^2 + 205^2 \; = \; 271^2 + 103^2 \; = \; 84050$
$133^2 + 156^2 + 205^2 \; = \; 289^2 + 23^2 \; = \; 84050$
$21^2 + 220^2 + 221^2 \; = \; 241^2 + 199^2 \; = \; 97682$
$140^2 + 171^2 + 221^2 \; = \; 311^2 + 31^2 \; = \; 97682$
$60^2 + 221^2 + 229^2 \; = \; 281^2 + 161^2 \; = \; 104882$
$105^2 + 208^2 + 233^2 \; = \; 313^2 + 103^2 \; = \; 108578$
$120^2 + 209^2 + 241^2 \; = \; 329^2 + 89^2 \; = \; 116162$
$32^2 + 255^2 + 257^2 \; = \; 287^2 + 223^2 \; = \; 132098$
$23^2 + 264^2 + 265^2 \; = \; 287^2 + 241^2 \; = \; 140450$
$96^2 + 247^2 + 265^2 \; = \; 343^2 + 151^2 \; = \; 140450$
$69^2 + 260^2 + 269^2 \; = \; 329^2 + 191^2 \; = \; 144722$
$115^2 + 252^2 + 277^2 \; = \; 367^2 + 137^2 \; = \; 153458$
$160^2 + 231^2 + 281^2 \; = \; 391^2 + 71^2 \; = \; 157922$
$161^2 + 240^2 + 289^2 \; = \; 401^2 + 79^2 \; = \; 167042$
$68^2 + 285^2 + 293^2 \; = \; 353^2 + 217^2 \; = \; 171698$