# Classical geometry statement in modern terminology

Given two line segments $\overline{AB}$ and $\overline{CD}$, it's always possible to find a third line segment whose length divides evenly into the first two. In modern terminology, if we assign $x = \overline{AB}$ and $y = \overline{CD}$, than the above statement is equivalent to asserting that $x = ay$, where $a \in \mathbb{Q}$.

I'm having difficulty understanding why these two statements are equivalent, mainly because I find the phrasing of the first sentence confusing. If $\overline{CD}$ is a rational multiple of $\overline{AB}$, then why can we always find a third line segment that "evenly" divides into them?

If $a = \frac p q$, then $x$ and $y$ are both integral multiples of $\frac y q$ by multplying by $p$ and by $q$ respectively. So here, $\frac p q$ is the length of the third length.