Constraints: $f^2<g^2<a^2$
The equations are
$2g^2−f^2=A$
$2a^2−f^2=B$
$g^2−f^2+a^2=C$
$g^2−2f^2+2a^2=D$
$2g^2−3f^2+2a^2=E$
$2g^2−2f^2+a^2=F$
Find the values of $a$, $f$ and $g$ such that the results of all above equations (capital lettered variables) are a perfect square.
How do I go about solving this? How many solutions are possible? What could be one trivial solution?
Please provide some useful insight or point to a learning resource before downvoting. Any edits are welcome as well.
Have you tried, for example, do something like
$$(I)\ \ g^2-f^2+a^2=C $$ $$(II)\ \ g^2-2f^2+2a^2= D $$ so $$-(I)+(II) = -f+a^2 = D-C$$
and use this fact to solve the other expressions, such as $$2a^2-f^2 = B = a^2 +a^2-f^2 =a^2+ D-C$$ and so $$a^2=B+C-D$$