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 $a : c = c : b$.
In the words of Euclid:
- To two given straight lines to find a mean proportional.
(The Elements: Book $\text{VI}$: 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 proof is Proposition $13$ of Book $\text{VI}$ of Euclid's The Elements.
Also see Proposition $2$ of Book $\text{II} $: 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.
Sources
- 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 2 (2nd ed.) ... (previous) ... (next): Book $\text{VI}$. Propositions