Why does the F-ratio follow an F-distribution when the null hypothesis of equal grouped means is satisfied?

I'm really stuck on this.

I know that an F distribution is:

$\displaystyle \displaystyle \frac{\frac{\chi^{2}_1}{df_1}}{\frac{\chi^{2}_2}{d f_2}}$

And the F-Ratio is:

MST= mean squares for treatment

MSE= mean squares for error

$\displaystyle \frac{MST}{MSE}$

From here I'm totally stuck.

I think it all boils down to me not understanding why

$\displaystyle \display \frac{SST}{\sigma^2}~{\chi^2}_{a-1}$

Only when all of the group means equal each other.

It seems that

$\displaystyle \frac{SSE}{\sigma^2}~\chi^{2}_{N-a}$

is always a chi squared distribution.