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I tried to prove Routh's theorem from geometry and while solving it I had to simpily $$1-\frac{s}{st+s+1}-\frac{t}{rt+t+1}-\frac{r}{rs+r+1}$$ to $$\frac{(rst-1)^2}{(st+s+1)(rt+t+1)(rs+r+1)}$$ I managed to do it by multiplying everything out and cancelling many terms but is there any "clever" way to see the identity?

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