This is not yet an answer, but in terms of $\beta$ functions, one has
$\beta(a,1-a-b)+\beta(b,1-a-b)+\beta(a,b)=\displaystyle\frac{\beta\left(\frac{b}{2},\frac{1}{2}\right)}{\beta\left(\frac{1-a}{2},\frac{a+b}{2}\right)}$
(use the fact that $\Gamma(1/2)=\sqrt{\pi})$.
Now, i think one can use the additive properties that beta functions enjoy such as $\beta(a,b)=\beta(a+1,b)+\beta(a,b+1)$. hmm....