## {(a^2 b^2) + n^2(a^2 + b^2), (a^2 b^2) + n^2 c^2}

Find three squares such that the product of any two added to the product of a given
square   $n^2$   by either the sum of those two or the remaining one gives a square

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

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

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

are all squares.

Here’s one example,

$a^2 = 5^2$,     $b^2 = 8^2$,     $c^2 = 14^2$,     $n = 3$

$(5^2\times 8^2) \; + \; 3^2 \,(5^2 + 8^2) \; = \; 49^2$
$(5^2\times 8^2) \; + \; (3^2\times 14^2) \; = \; 58^2$

$(5^2\times 14^2) \; + \; 3^2 \,(5^2 + 14^2) \; = \; 83^2$
$(5^2\times 14^2) \; + \; (3^2\times 8^2) \; = \; 74^2$

$(8^2\times 14^2) \; + \; 3^2 \,(8^2 + 14^2) \; = \; 122^2$
$(8^2\times 14^2) \; + \; (3^2\times 5^2) \; = \; 113^2$