Studying Feynman diagrams in 1+1 dimensions, I found the following sum over cyclic permutations
$f(a,b,c,d,e,f)=\sum_{cycl. \{a,b,c\}}\sum_{cycl. \{d,e,f\}}\frac{abd}{(a+b)(a-d)(b-d)}$.
Then you can check with Mathematica (already not totally trivial) that the if $$a+b+c=d+e+f~~~(1)$$ and $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{d}+\frac{1}{e}+\frac{1}{f}~~~(2)$$ the function $f(a,b,c,d,e,f)$ becomes $f(a,b,c,d,e,f)=1+\frac{abc}{(a+b)(a+c)(b+c)}$. How can I prove this identity without resorting to brute force?
The relations $(1,2)$ are energy and momentum conservation. For more information about the QFT background see https://arxiv.org/abs/hep-th/9810026.