Construction of Equal Straight Lines from Unequal

Theorem

Given two unequal straight line segments, it is possible to cut off from the greater a straight line segment equal to the lesser.

Construction

Let $AB$ and $c$ be the given straight line segments.

Let $AB$ be the greater of them.

At point $A$, we place $AD$ equal to $c$.

We construct a circle $DEF$ with center $A$ and radius $AD$.

The straight line segment $AE$ is the required line.

Proof

As $A$ is the center of circle $DEF$, it follows from Book I Definition 15: Circle that $AE = AD$.

But $c$ is also equal to $AD$.

So, as $c = AD$ and $AD = AE$, it follows from Common Notion 1 that $AE = c$.

Therefore, given the two straight line segments $AB$ and $c$, from the greater of these $AB$, a length $AE$ has been cut off equal to the lesser $c$.

$\blacksquare$

Historical Note

This is Proposition 3 of Book I of Euclid's The Elements.