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 $a : c = c : b$.

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

 * Euclid-VI-13.png

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 $BC$.