Finding integer solutions to $ \frac{a^3+b^3}{a^3+c^3} = \frac{a+b}{a+c} $ I was browsing through facebook and came across this image: 
I was wondering if we can find more examples where this happens?
I guess this reduces to finding integer solutions for the equation 
$$ \frac{a^3+b^3}{a^3+c^3} = \frac{a+b}{a+c} $$ for integers a,b,c
Or can we even further extend to when they are all distinct that is finding solutions to 
$$ \frac{a^3+b^3}{c^3+d^3} = \frac{a+b}{c+d} $$ for integers a,b,c
I don't really have that much knowledge in the number theory area so I have come here

$$\frac{a^3+b^3}{a^3+c^3}=\frac{(a+b)(a^2-ab+b^2)}{(a+c)(a^2-ac+c^2)}=\frac{a+b}{a+c}$$
If $a+c \neq 0$ and $a+b \neq 0,$ then $$a^2-ab+b^2=a^2-ac+c^2,$$namely $$(b+c-a)(b-c)=0.$$
If $b=c$, the case is trivial. If $b \neq c$, then $$b+c=a.$$
