Relation of Ratios to Products

Theorem

 * If four (natural) numbers be proportional, the number produced from the first and fourth will be equal to the number produced from the second and third; and, if the number produced from the first and fourth be equal to the number produced from the second and third, the four numbers will be proportional.

That is:
 * $a : b = c : d \iff ad = bc$

Proof
Let $A, B, C, D$ be four (natural) numbers in proportion, so that $A : B = C : D$.

Let $A \times D = E$ and $B \times C = F$.

We need to show that $E = F$.


 * Euclid-VII-19.png

Let $A \times C = G$.

Then $A \times C = G$ and $A \times D = E$.

So from :
 * $C : D = G : E$

But we have:
 * $C : D = A : B$
 * $A : B = G : E$
 * $A \times C = G$
 * $B \times C = F$

Then from :
 * $A : B = G : F$

Further, we have that:
 * $A : B = G : E$

Thus:
 * $G : F = G : E$

So from :
 * $E = F$

Now suppose that $E = F$.

We need to show that:
 * $A : B = C : D$

Using the same construction, from :
 * $G : E = G : F$

But from :
 * $G : E = C : D$

Then from :
 * $G : F = A : B$

So:
 * $A : B = C : D$