Construction of Third Proportional Straight Line

Theorem
Given any two straight lines of given length $a$ and $b$, it is possible to construct a third straight line of length $c$ such that $a : b = b : c$.

Construction
Let $AB, AC$ be the two given straight lines.

Let them be placed to contain any angle.

Let $AB$ be produced to $D$, and $AC$ be produced to $E$.

Let $BD$ be constructed equal to $AC$.

Join $BC$ and construct $DE$ parallel to $BC$.

Then $CE$ is the required third proportional line.

Proof

 * Euclid-VI-11.png

We have that:
 * $BC \parallel DE$

So from Parallel Line in Triangle Cuts Sides Proportionally:
 * $AB : BD = AC : CE$

But $BD = AC$ and so:
 * $AB : AC = AC : CE$

as required.