In linear regression I have come across a delightful result that if we fit the model

$$E[Y] = \beta_1 X_1 + \beta_2 X_2 + c,$$

then, if we standardize and centre the $Y$, $X_1$ and $X_2$ data,

$$R^2 = \mathrm{Cor}(Y,X_1) \beta_1 + \mathrm{Cor}(Y, X_2) \beta_2.$$

This feels to me like a 2 variable version of $R^2 = \mathrm{Cor}(Y,X)^2$ for $y=mx+c$ regression, which is pleasing.

But the only proof I know is not in anyway constructive or insightful (see below), and yet to look at it it feels like it should be readily understandable.

Example thoughts: 

* The $\beta_1$ and $\beta_2$ parameters give us the 'proportion' of $X_1$ and $X_2$ in $Y$, and so we are taking respective proportions of their correlations...
* The $\beta$s are partial correlations, $R^2$ is the squared multiple correlation... correlations multiplied by partial correlations...
* If we orthogonalize first then the $\beta$s will be $\mathrm{Cov}/\mathrm{Var}$... does this result make some geometric sense?

None of these threads seem to lead anywhere for me.  Can anyone provide a clear explanation of how to understand this result.

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**Unsatisfying Proof**

\begin{equation}
R^2 = \frac{SS_{reg}}{SS_{Tot}} = SS_{reg} = \langle(\beta_1 X_1 + \beta_2 X_2)^2\rangle
 \\= \langle\beta_1^2 X_1^2\rangle + \langle\beta_2^2 X_2^2\rangle + 2\langle\beta_1\beta_2X_1X_2\rangle
\end{equation}

and

\begin{equation}
\mathrm{Cor}(Y,X_1) \beta_1 + \mathrm{Cor}(Y, X_2) \beta_2 = \langle YX_1\rangle\beta_1 + \langle Y X_2\rangle \beta_2\\
=\langle \beta_1 X_1^2 + \beta_2 X_1 X_2\rangle \beta_1 + \langle \beta_1 X_1 X_2 + \beta_2 X_2^2\rangle \beta_2\\
=\langle \beta_1^2 X_1^2\rangle + \langle \beta_2^2 X_2^2 \rangle + 2\langle \beta_1 \beta_2 X_1 X_2\rangle
\end{equation}

QED.