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Proof: By Euclid

(related to Proposition: Prop. 9.30: Odd Divisor of Even Number Also Divides Its Half)

• For let the odd number $A$ measure the even (number) $B$.
• I say that ($A$) will also measure (one) half of ($B$).

• 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)

1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"