How would Euclid have proved that $ a \times b = ((ab)/c) \times c$?

Question

The following statement seems too obvious to prove or even to mention:

That the area of a rectangle with side lengths $a$ and $b$ equals the area of a rectangle with side lengths $(ab)/c$ and $c$ for arbitrary $c>0$, i.e.

$$ a \times b = ((ab)/c) \times c$$

(Note that $(ab)/c$ is a shorthand for $a/(c/b)$.)

I wonder how this statement would have been formulated by Euclid and how he would have proved it (as a geometrical statement with geometrical methods).

I guess there is no proposition in Euclid's Elements that says exactly this, but maybe there is one which says essentially the same - or there is an easy proof starting from theorems already proved by Euclid.

This is how how I would formulate the statement:

Consider two figures constructed like this:

and like that:

Then they are equal (in size).

Answer 1

Two thoughts:

Euclid's Proposition II.14 is "to construct a square equal to a given rectilinear figure."

Most of the proof consists of showing that any rectangle is equal in area to a square. This follows from the Pythagorean Theorem and an earlier result in the chapter (Proposition II.5) which decomposes a rectangle as the difference of two squares.

It immediately follows that given any rectangle $R$ and any segment $AB$, you can construct another rectangle with side $AB$ and area equal to $R$. I'm not sure how easy it is to see that this rectangle must be the same one as in your construction, though.

Alternately, you could check out the treatment of similarity and area in Hilbert (chapters III and IV). Euclid would have been comfortable with all his constructions (if perhaps not with their interpretations), and he makes extensive use of exactly the sort of parallel-line construction you are using to define multiplication of line segments. I don't think he proves exactly your theorem, but he proves things which I'm sure are equivalent to it with a little work. (In particular, he rigorously proves that area is well-defined, which is sufficient to fill the gap I mentioned in the proof following Euclid — though possibly overkill.)