Proposition: 7.09: Alternate Ratios of Equal Fractions
Euclid's Formulation
If a number is part of a number, and another (number) is the same part of another, also, alternately, which(ever) part, or parts, the first (number) is of the third, the second (number) will also be the same part, or the same parts, of the fourth.
 For let a number $A$ be part of a number $BC$, and another (number) $D$ (be) the same part of another $EF$ that $A$ (is) of $BC$.
 I say that, also, alternately, which(ever) part, or parts, $A$ is of $D$, $BC$ is also the same part, or parts, of $EF$.
Modern Formulation
In modern notation, this proposition states that if $A=\frac{BC}{n}$ and $D=\frac{EF}{n}$ for some integers $n \ge 1$ and $0 < BC < EF,$ then, using the division with quotient and remainder,
$$\begin{array}{rclccl}
D&=&m\cdot A&+&r&0\le r < A\\
&\Updownarrow&\\
EF&=&m\cdot BC&+&nr&0\le nr < BC.
\end{array}$$
Table of Contents
Proofs: 1
Mentioned in:
Proofs: 1 2
Propositions: 3
Thank you to the contributors under CC BYSA 4.0!
 Github:

 nonGithub:
 @Fitzpatrick
References
Adapted from (subject to copyright, with kind permission)
 Fitzpatrick, Richard: Euclid's "Elements of Geometry"
Adapted from CC BYSA 3.0 Sources:
 Prime.mover and others: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016