In my research, I came up with this equation which consists of 4 variables (t,m,g,h).
$$4tm(t-m)(t+m)=h(\sqrt{3}g-h)(\sqrt{3}g+h)$$
Initially, I attempted in equating each product on LHS and RHS, as follows
(i) $4tm=h$, $t-m=\sqrt{3}g-h$, $t+m=\sqrt{3}g+h$
(ii) $4tm=h$, $t-m=\sqrt{3}g+h$, $t+m=\sqrt{3}g-h$
...
(vi) $4tm=\sqrt{3}g+h$, $t-m=\sqrt{3}g-h$, $t+m=h$
Under all these assumptions, I manage to proof contradictions that no such solutions exist.
Later on I was thinking that these assumptions might not be correct, because 8*6*4=2*3*32, in which no factors on both sides of the equality are equal. These are only correct iff both sides of the equality are product of different primes. Hence, to apply (i) to (vi), I should first of all establish that the factors on both sides of this equality are primes (no idea yet on how to do that, perhaps work with the prime power decomposition). Can someone help me shed some light on how to tackle this problem. I need to figure out if there exist solutions or not to this equation.