Rectangles Contained by Three Proportional Straight Lines
Theorem
In the words of Euclid:
- If three straight lines be proportional, the rectangle contained by the extremes is equal to the square on the mean; and, if the rectangle contained by the extremes is equal to the square on the mean, the three straight lines will be proportional.
(The Elements: Book $\text{VI}$: Proposition $17$)
Proof
Let the three straight lines $A, B, C$ be proportional, that is:
- $A : B = B : C$
Then we need to show that the rectangle contained by $A$ and $C$ equals the square on $B$.
Let $D = B$.
Then $A : B = D : C$
By Rectangles Contained by Proportional Straight Lines, the rectangle contained by $A$ and $C$ equals the rectangle contained by $B$ and $D$.
But as $B = D$, the rectangle contained by $B$ and $D$ equals the square on $B$.
So the rectangle contained by $A$ and $C$ equals the square on $B$.
$\Box$
Now let the rectangle contained by $A$ and $C$ be equal to the square on $B$.
Using the same construction, the rectangle contained by $A$ and $C$ equals the rectangle contained by $B$ and $D$ because $B = D$.
So by Rectangles Contained by Proportional Straight Lines:
- $A : B = D : C$
But as $B = D$:
- $A : B = B : C$
$\blacksquare$
Historical Note
This proof is Proposition $17$ of Book $\text{VI}$ of Euclid's The Elements.
This is, of course, just a special case of Proposition $16$ of Book $\text{VI} $: Rectangles Contained by Proportional Straight Lines.
Sources
- 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 2 (2nd ed.) ... (previous) ... (next): Book $\text{VI}$. Propositions