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 Definition I-15 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 $$c$$.

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