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Proof: By Euclid

(related to Proposition: Prop. 10.024: Rectangle Contained by Medial Straight Lines Commensurable in Length is Medial)

For let the rectangle $AC$ be contained by the medial straight lines $AB$ and $BC$ (which are) commensurable in length.
I say that $AC$ is medial.

fig024e

For let the square $AD$ have been described on $AB$.
$AD$ is thus medial [see previous footnote].
And since $AB$ is commensurable in length with $BC$, and $AB$ (is) equal to $BD$, $DB$ is thus also commensurable in length with $BC$.
Hence, $DA$ is also commensurable with $AC$ [Prop. 6.1], [Prop. 10.11].
And $DA$ (is) medial.
Thus, $AC$ (is) also medial [Prop. 10.23 corr.] .
(Which is) the very thing it was required to show.
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References

Adapted from (subject to copyright, with kind permission)

Fitzpatrick, Richard: Euclid's "Elements of Geometry"