Let it be four odd positive integers $a,b,c,d$. Is it possible the following relationship?
$$abcd = 2abc+bcd+cda+dab$$
Operating,
$$abc(d-2)=d(bc+ca+ab)$$
$$\frac{d}{d-2}=\frac{abc}{ab+bc+ca}$$
I got stuck at this point. Clearly, $d$ and $d-2$ are relatively primes, as they are odd, but I am not sure that this is substantial enough to get any conclusion.
Any hint / comment on how to follow would be welcomed!