I came across the following in some old class notes of mine:
if $\chi_{v_{1}}^{2}$ is independent of $\chi_{v_{2}}^{2}$ then $\frac{\chi_{v_{1}}^{2}}{\chi_{v_{1}}^{2}+\chi_{v_{2}}^{2}}\backsim Beta\left(\alpha=\frac{v_{1}}{2},\beta=\frac{v_{2}}{2}\right)$,
but there was no proof presented. I tried to prove this on my own, but I'm getting stuck when trying to get the $(1-x)^{\beta-1}$ term from the $Beta(\alpha, \beta)$ distribution to appear. My attempt is below, but I'm getting stuck, so I'd appreciate it if anyone could help me figure out where I've gone wrong or perhaps next steps to take to prove this.
\begin{eqnarray*} \frac{\chi_{v_{1}}^{2}}{\chi_{v_{1}}^{2}+\chi_{v_{2}}^{2}} & = & \frac{\chi_{v_{1}}^{2}}{\chi_{v_{1}+v_{2}}^{2}}&(\text{independence})\\ & = & \frac{x^{(v_{1}/2)-1}e^{x/2}}{\Gamma\left(\frac{v_{1}}{2}\right)2^{\left(v_{1}/2\right)}}\cdot\frac{\Gamma\left(\frac{v_{1}}{2}+\frac{v_{2}}{2}\right)2^{\left[\left(v_{1}+v_{2}\right)/2\right]}}{x^{(v_{1}+v_{2}/2)-1}e^{x/2}}\\ & = & \frac{x^{\alpha-1}}{\Gamma\left(\alpha\right)2^{\alpha}}\cdot\frac{\Gamma\left(\alpha+\beta\right)2^{(\alpha+\beta)}}{x^{\alpha+\beta-1}}&(\text{letting } \frac{v_{1}}{2}=\alpha ,\frac{v_{2}}{2}=\beta )\\ & = & \frac{\Gamma\left(\alpha+\beta\right)}{\Gamma\left(\alpha\right)}2^{\beta}x^{\alpha-1}x^{-\alpha-\beta+1}\\ & = & \frac{\Gamma\left(\alpha+\beta\right)}{\Gamma\left(\alpha\right)\Gamma\left(\beta\right)}\Gamma\left(\beta\right)2^{\beta}x^{\alpha-1}x^{-\alpha-\beta+1}\\ & = & \frac{1}{Beta(\alpha,\beta)}x^{\alpha-1}x^{-\alpha-\beta+1}\\ & = & \frac{1}{Beta(\alpha,\beta)}x^{\alpha-1}\underset{ \text{need } (1-x)^{\beta-1} \text{term from this?}}{\underbrace{\Gamma\left(\beta\right)2^{\beta}x^{-\alpha-\beta+1}}} \end{eqnarray*}
So, it seems like I should be able to make the terms gathered above the underbrace to form the $(1-x)^{\beta-1}$ term, but I can't see how to do this. It's entirely possible I've gone wrong someplace as well. I'd appreciate any help.