In Euclid's day, the modern notion of real number did not exist; Euclid did not believe that the length of a line segment was a quantity measurable by number. But he did think it made sense to talk about the ratio of two lengths. In fact, he devotes Book V of his Elements to the study of such ratios, using the so-called Eudoxian theory of proportions. Here's how it works.
Let $w$ and $x$ be two magnitudes of the same kind (for instance two length), and let $y$ and $z$ be two magnitudes of the same kind (for instance two areas). Then according to Euclid, the ratio of $w$ to $x$ is said to be equal to the ratio of $y$ to $z$ if for all positive integers $m$ and $n$, if $nw$ is greater, equal, or less than $mx$, then $ny$ is greater, equal, or less than $mz$, respectively. Or to put it in modern language, $w/x = y/z$ if the same rational numbers $m/n$ are less than both, the same rational numbers are equal to both, and the same rational numbers are greater than both.
In other words, a ratio is defined by the classes of rational numbers which are less than, equal to, and greater than it. If you've studied real analysis; this should look familiar to you: it is how the real number system is constructed using Dedekind cuts! In fact, Dedekind took the Eudoxian theory of proportions in Euclid's Book V as the inspiration for his Dedekind cut construction. So to sum up, while Euclid wouldn't have thought of them as numbers, his notion of "ratios" basically corresponds to our notion of "positive real numbers".
Now with that background, in this question I wanted to try to prove that real number multiplication is commutative, but it turned out that Euclid had beat me to the punch. Now I'd like to prove that multiplication of real numbers distributes over addition. First let me explain how the sum and product of two ratios is defined. We say that the sum of $w/x$ and $y/z$ is equal to $u/v$ if there exist magnitudes $r,s,$ and $t$ such that $w/x = r/t$, $y/z = s/t$, and $(r+s)/t = u/v$. And we say that the product of $w/x$ and $y/z$ is equal to $u/v$ if there exist magnitudes $r,s,$ and $t$ such that $w/x = r/s$, $y/z = s/t$, and $r/t = u/v$.
So in order to establish that multiplication distributes over addition, we need to prove the following:
Suppose that the product of $a/b$ and $c/e$ is $f/h$, and the product of $a/b$ and $d/e$ is $g/h$. Then the product of $a/b$ and $(c+d)/e$ is $(f+g)/h$
So how would I go about proving that? Euclid's Book V contains a lot of theorems about ratios that are potentially relevant, but I'm not sure how to proceed.
Note that there are other kinds of distributive properties proved in Euclid's Elements, including this one, this one, and this one, but they're not relevant here.
Just working with lengths and areas, not ratios, the ordinary distributive law is proved in book II, proposition 1: http://aleph0.clarku.edu/~djoyce/elements/bookII/propII1.html
Searching a little more, there is book VII, proposition 6: If a number be parts of a number, and another be the same parts of another, the sum will also be the same parts of the sum that the one is of the one. Here: https://proofwiki.org/wiki/Multiples_of_Divisors_obey_Distributive_Law/Proof_1
Note: both of these were found with a Google search of "euclid distributive law".