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I have seen this equality many times in books but I never found an explanatory derivation.

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    $\begingroup$ It has appeared a very great many times on this site. The main difficulty is that it's almost impossible to search for the answers! $\endgroup$ – whuber Jan 5 '20 at 21:48
  • $\begingroup$ It is quite difficult to search equations in a text-match-based searcher $\endgroup$ – E. Williams Jan 5 '20 at 22:07
  • $\begingroup$ Agreed! The duplicates I found are only the tip of the iceberg; I searched stats.stackexchange.com/search?q=regression+orthogonal+residual. Most of the demonstrations are absolutely identical to the answer you received (right down to the variable names), but the threads mention none of these terms. $\endgroup$ – whuber Jan 5 '20 at 22:20
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$$\sum{(x_i-\overline{x})^2} = \sum{(x_i-\overline{x})x_i} - \sum{(x_i-\overline{x})\overline{x}} = \sum{(x_i-\overline{x})x_i} - \overline{x}\sum{x_i} + n\overline{x}^2 = \sum{(x_i-\overline{x})x_i} - \overline{x}n\overline{x} + n\overline{x}^2 = \sum{(x_i-\overline{x})x_i}$$

$\sum{x_i} = n\overline{x}$

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