Independence of a linear combination of Normal Random Variables

Question

I would like to prove the following:

I have that $X_1, X_2$ are 2 random variables, each independently following a $N(\theta,1)$ distribution.

I firstly need to show that: $ X_1 - \bar{X}$ is independent of $\bar{X}$

And from this derive the distribution of $X_1$ given $\bar{X}$. Is there a neat/easy way of doing this?

Note in the above, $\bar{X}$ denotes the mean of the 2 random variables, and $\theta$ is unknown.

Thanks in advance!