Proof: By Euclid
(related to Lemma: Lem. 10.021: Medial is Irrational)
 For let the square $DF$ have been described on $FE$.
 And let $GD$ have been completed.
 Therefore, since as $FE$ is to $EG$, so $FD$ (is) to $DG$ [Prop. 6.1], and $FD$ is the (square) on $FE$, and $DG$ the (rectangle contained) by $DE$ and $EG$  that is to say, the (rectangle contained) by $FE$ and $EG$  thus as $FE$ is to $EG$, so the (square) on $FE$ (is) to the (rectangle contained) by $FE$ and $EG$.
 And also, similarly, as the (rectangle contained) by $GE$ and $EF$ is to the (square on) $EF$  that is to say, as $GD$ (is) to $FD$  so $GE$ (is) to $EF$.
 (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"