Rectangles Contained by Proportional Straight Lines
Theorem
In the words of Euclid:
- If four straight lines be proportional, the rectangle contained by the extremes is equal to the rectangle contained by the means; and, if the rectangle contained by the extremes is equal to the rectangle contained by the means, the four lines will be proportional.
(The Elements: Book $\text{VI}$: Proposition $16$)
Note: in the above, equal is to be taken to mean of equal area.
Proof
Let the four straight lines $AB, CD, E, F$ be proportional, that is, $AB : CD = E : F$.
What we need to show is that the rectangle contained by $AB$ and $F$ is equal in area to the rectangle contained by $CD$ and $E$.
Let $AG, CH$ be drawn perpendicular to $AB$ and $CD$.
Let $AG = F$, $CH = E$.
Complete the parallelograms $BG$ and $DH$.
We have that $AB : CD = E : F$, while $E = CH$ and $F = AG$.
So in $\Box BG$ and $\Box DH$ the sides about the equal angles are reciprocally proportional.
But from Sides of Equal and Equiangular Parallelograms are Reciprocally Proportional:
- $\Box BG = \Box DH$ (in area).
We also have that:
- $\Box BG$ is the rectangle contained by $AB$ and $F$
- $\Box DH$ is the rectangle contained by $CD$ and $E$
Hence the result.
$\Box$
Now suppose that the rectangle contained by $AB$ and $F$ is equal in area to the rectangle contained by $CD$ and $E$.
We use the same construction, and note that $\Box BG = \Box DH$ (in area).
But they are equiangular, as all angles are equal to a right angle.
So from Sides of Equal and Equiangular Parallelograms are Reciprocally Proportional:
- $AB : CD = CH : AG$
But $E = CH$ and $F = AG$.
So:
- $AB : CD = E : F$
$\blacksquare$
Historical Note
This proof is Proposition $16$ of Book $\text{VI}$ of Euclid's The Elements.
It is a special case of Proposition $14$ of Book $\text{VI} $: Sides of Equal and Equiangular Parallelograms are Reciprocally Proportional.
Sources
- 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 2 (2nd ed.) ... (previous) ... (next): Book $\text{VI}$. Propositions