Proof: By Euclid
(related to Proposition: Prop. 9.30: Odd Divisor of Even Number Also Divides Its Half)
 For since $A$ measures $B$, let it measure it according to $C$.
 I say that $C$ is not odd.
 For, if possible, let it be (odd).
 And since $A$ measures $B$ according to $C$, $A$ has thus made $B$ (by) multiplying $C$.
 Thus, $B$ is composed out of odd numbers, (and) the multitude of them is odd.
 $B$ is thus odd [Prop. 9.23].
 The very thing (is) absurd.
 For ($B$) was assumed (to be) even.
 Thus, $C$ is not odd.
 Thus, $C$ is even.
 Hence, $A$ measures $B$ an even number of times.
 So, on account of this, ($A$) will also measure (one) half of ($B$).
 (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"