Proof: By Euclid
(related to Proposition: Prop. 10.086: Construction of Second Apotome)
 Let the rational (straight line) $A$, and $GC$ (which is) commensurable in length with $A$, be laid down.
 Thus, $GC$ is a rational (straight line).
 And let the two square numbers $DE$ and $EF$ be laid down, and let their difference $DF$ be not square [Prop. 10.28 lem. I] .
 And let it have been contrived that as $FD$ (is) to $DE$, so the square on $CG$ (is) to the square on $GB$ [Prop. 10.6 corr.] .
 Thus, the square on $CG$ is commensurable with the square on $GB$ [Prop. 10.6].
 And the (square) on $CG$ (is) rational.
 Thus, the (square) on $GB$ [is] also rational.
 Thus, $BG$ is a rational (straight line).
 And since the square on $GC$ does not have to the (square) on $GB$ the ratio which (some) square number (has) to (some) square number, $CG$ is incommensurable in length with $GB$ [Prop. 10.9].
 And they are both rational (straight lines).
 Thus, $CG$ and $GB$ are rational (straight lines which are) commensurable in square only.
 Thus, $BC$ is an apotome [Prop. 10.73].

So, I say that it is also a second (apotome) .

For let the (square) on $H$ be that (area) by which the (square) on $BG$ is greater than the (square) on $GC$ [Prop. 10.13 lem.] .
 Therefore, since as the (square) on $BG$ is to the (square) on $GC$, so the number $ED$ (is) to the number $DF$, thus, also, via convertion, as the (square) on $BG$ is to the (square) on $H$, so $DE$ (is) to $EF$ [Prop. 5.19 corr.] 2.
 And $DE$ and $EF$ are each square (numbers).
 Thus, the (square) on $BG$ has to the (square) on $H$ the ratio which (some) square number (has) to (some) square number.
 Thus, $BG$ is commensurable in length with $H$ [Prop. 10.9].
 And the square on $BG$ is greater than (the square on) $GC$ by the (square) on $H$.
 Thus, the square on $BG$ is greater than (the square on) $GC$ by the (square) on (some straight line) commensurable in length with ($BG$).
 And the attachment $CG$ is commensurable (in length) with the (prevously) laid down rational (straight line) $A$.
 Thus, $BC$ is a second apotome [Def. 10.12] .
 Thus, the second apotome $BC$ has been found.
 (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"