Rectangles Contained by Three Proportional Straight Lines

Theorem

 * 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.

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 $B$.


 * Euclid-VI-17.png

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 $B$.

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