Construction of Mean Proportional

Theorem

Given any two straight lines of length $a$ and $b$ it is possible to find a straight line of length $c$ such that $a : c = c : b$.

As Euclid defined it:

To two given straight lines to find a mean proportional.

(The Elements: Book VI: Proposition $13$)

Construction

Let $AB$ and $BC$ be the two given straight lines.

We require to find a mean proportional to $AB, BC$.

Let $AB$ and $BC$ be placed in a straight line and let the semicircle $ADC$ be placed on $AC$.

Let $BD$ be drawn perpendicular to $AC$.

Then $BD$ is the required mean proportional.

Proof

From Relative Sizes of Angles in Segments, $\angle ADC$ is a right angle.

So from the porism to Perpendicular in Right-Angled Triangle makes two Similar Triangles, $DB$ is the mean proportional between $AB$ and $BC$.

$\blacksquare$

Historical Note

This is Proposition 13 of Book VI of Euclid's The Elements.

Also see Book II Proposition 2: Construction of Square Equal to Given Polygon for what amounts to an application of this technique.
Note that nowhere in The Elements is the term mean proportional specifically defined.