Proof: By Euclid
(related to Proposition: Prop. 13.06: Segments of Rational Straight Line cut in Extreme and Mean Ratio are Apotome)
 Let $AB$ be a rational straight line cut in extreme and mean ratio at $C$, and let $AC$ be the greater piece.

I say that $AC$ and $CB$ is each that irrational (straight line) called an apotome.

For let $BA$ have been produced, and let $AD$ be made (equal) to half of $BA$.
 Therefore, since the straight line $AB$ has been cut in extreme and mean ratio at $C$, and $AD$, which is half of $AB$, has been added to the greater piece $AC$, the (square) on $CD$ is thus five times the (square) on $DA$ [Prop. 13.1].
 Thus, the (square) on $CD$ has to the (square) on $DA$ the ratio which a number (has) to a number.
 The (square) on $CD$ (is) thus commensurable with the (square) on on $DA$ [Prop. 10.6].
 And the (square) on $DA$ (is) rational.
 For $DA$ [is] [rational]bookofproofs$2083, being half of $AB$, which is rational.
 Thus, the (square) on $CD$ (is) also rational [Def. 10.4] .
 Thus, $CD$ is also rational.
 And since the (square) on $CD$ does not have to the (square) on $DA$ the ratio which a square number (has) to a square number, $CD$ (is) thus incommensurable in length with $DA$ [Prop. 10.9].
 Thus, $CD$ and $DA$ are rational (straight lines which are) commensurable in square only.
 Thus, $AC$ is an apotome [Prop. 10.73].
 Again, since $AB$ has been cut in extreme and mean ratio, and $AC$ is the greater piece, the (rectangle contained) by $AB$ and $BC$ is thus equal to the (square) on $AC$ [Def. 6.3] , [Prop. 6.17].
 Thus, the (square) on the apotome $AC$, applied to the rational (straight line) $AB$, makes $BC$ as width.
 And the (square) on an apotome, applied to a rational (straight line), makes a first apotome as width [Prop. 10.97].
 Thus, $CB$ is a first apotome.
 And $CA$ was also shown (to be) an apotome.
 Thus, if a rational straight line is cut in extreme and mean ratio then each of the pieces is that irrational (straight line) called an apotome.
∎
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References
Adapted from (subject to copyright, with kind permission)
 Fitzpatrick, Richard: Euclid's "Elements of Geometry"