Lemma: Lem. 10.053: Construction of Rectangle with Area in Mean Proportion to two Square Areas
(Lemma to Proposition 53 from Book 10 of Euclid's “Elements”)
Let $AB$ and $BC$ be two squares, and let them be laid down such that $DB$ is straighton to $BE$. $FB$ is, thus, also straighton to $BG$. And let the parallelogram $AC$ have been completed. I say that $AC$ is a square, and that $DG$ is in mean proportion to $AB$ and $BC$, and, moreover, $DC$ is in mean proportion to $AC$ and $CB$.
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Adapted from (subject to copyright, with kind permission)
 Fitzpatrick, Richard: Euclid's "Elements of Geometry"