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....

This is not yet an answer, but in terms of $B$ (beta) functions, one has

$B(a,1-a-b)+B(b,1-a-b)+B(a,b)=\displaystyle\frac{B\left(\frac{b}{2},\frac{1}{2}\right)}{B\left(\frac{1-a}{2},\frac{a+b}{2}\right)}$

(use the fact that $\Gamma(1/2)=\sqrt{\pi})$.

Now, i think one can use the additive properties that beta functions enjoy such as $B(a,b)=B(a+1,b)+B(a,b+1)$. hmm....