If random variables $X \sim \mathrm{Gamma}(\frac{n}{2}, \frac{1}{2})$ and $Y \sim \mathrm{Gamma}(\frac{m}{2}, \frac{1}{2})$, where $m$ and $n$ are constants, why does $\frac{X}{X + Y} \sim \mathrm{Beta}(\frac{n}{2}, \frac{m}{2})$?
In general, can we just combine Gamma-distributed variables like this into Beta-distributed ones?