Proof: By Euclid
(related to Proposition: Prop. 10.033: Construction of Components of Major)
 And since $AB$ and $BC$ are [two] unequal straight lines, and the square on $AB$ is greater than (the square on) $BC$ by the (square) on (some straight line which is) incommensurable (in length) with ($AB$).
 And a parallelogram, equal to one quarter of the (square) on $BC$  that is to say, (equal) to the (square) on half of it  (and) falling short by a square figure, has been applied to $AB$, and makes the (rectangle contained) by $AEB$.
 $AE$ is thus incommensurable (in length) with $EB$ [Prop. 10.18].
 And as $AE$ is to $EB$, so the (rectangle contained) by $BA$ and $AE$ (is) to the (rectangle contained) by $AB$ and $BE$.
 And the (rectangle contained) by $BA$ and $AE$ (is) equal to the (square) on $AF$, and the (rectangle contained) by $AB$ and $BE$ to the (square) on $BF$ [Prop. 10.32 lem.] .
 The (square) on $AF$ is thus incommensurable with the (square) on on $FB$ [Prop. 10.11].
 Thus, $AF$ and $FB$ are incommensurable in square.
 And since $AB$ is rational, the (square) on $AB$ is also rational.
 Hence, the sum of the (squares) on $AF$ and $FB$ is also rational [Prop. 1.47].
 And, again, since the (rectangle contained) by $AE$ and $EB$ is equal to the (square) on $EF$, and the (rectangle contained) by $AE$ and $EB$ was assumed (to be) equal to the (square) on $BD$, $FE$ is thus equal to $BD$.
 Thus, $BC$ is double $FE$.
 And hence the (rectangle contained) by $AB$ and $BC$ is commensurable with the (rectangle contained) by $AB$ and $EF$ [Prop. 10.6].
 And the (rectangle contained) by $AB$ and $BC$ (is) medial [Prop. 10.21].
 Thus, the (rectangle contained) by $AB$ and $EF$ (is) also medial [Prop. 10.23 corr.] .
 And the (rectangle contained) by $AB$ and $EF$ (is) equal to the (rectangle contained) by $AF$ and $FB$ [Prop. 10.32 lem.] .
 Thus, the (rectangle contained) by $AF$ and $FB$ (is) also medial.
 And the sum of the squares on them was also shown (to be) rational.
 Thus, the two straight lines, $AF$ and $FB$, (which are) incommensurable in square, have been found, making the sum of the squares on them rational, and the (rectangle contained) by them medial.
 (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"
Footnotes