Rectangles Contained by Three Proportional Straight Lines

Theorem

As Euclid defined it:

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 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 is Proposition 17 of Book VI of Euclid's The Elements.

This is, of course, just a special case of Book VI Proposition 16: Rectangles Contained by Proportional Straight Lines.