Background
Let $X_1$, $X_2$, $Y_1$, $Y_2$ be independent random variables such that the following information is true:
$$\begin{array}{l}{X_{1}-X_{2} \sim N\left(0,2 \sigma_{x}^{2}\right), \quad Y_{1}-Y_{2} \sim N\left(0,2 \sigma_{y}^{2}\right)} \end{array}$$
Question
Why is it true that the above two random variables are related by the F-distribution, as below?
$$ \frac{\left(X_{1}-X_{2}\right)^{2} / \sigma_{x}^{2}}{\left(Y_{1}-Y_{2}\right)^{2} / \sigma_{y}^{2}}=\frac{\left[\left(X_{1}-X_{2}\right) /\left(\sqrt{2} \sigma_{x}\right)\right]^{2}}{\left[\left(Y_{1}-Y_{2}\right) /\left(\sqrt{2} \sigma_{y}\right)\right]^{2}} \sim F_{1,1} $$
The left- and right-side expressions don’t agree, so I assume that you want $(X_1-X_2)^2/(2\sigma_x^2)$ and similarly in the denominator. In that case the numerator and denominator are squares of independent standard normals, that is, $\chi^2$ variables with one degree of freedom. Such a ratio of $\chi^2$s gives rise to an $F_{1,1}$.