Construction of a Part of a Line

Theorem

From any given straight line, it is possible to cut off any specified part.

As Euclid defined it:

From a given straight line to cut off a prescribed part.

(The Elements: Book VI: Proposition $9$)

Construction

Let $AB$ be the given straight line.

From $A$ draw $AC$ making any angle with $AB$.

On $AC$, take any point $D$ and make $AC$ the same multiple of $AD$ that $AB$ is to be of the required part which is to be cut from it.

Join $BC$ and draw $DE$ parallel to it.

Then $AE$ is the required part of $AB$.

Proof

We have that $ED$ is parallel to one of the sides of $\triangle ABC$.

So from Parallel Line in Triangle Cuts Sides Proportionally, $CD : DA = BE : EA$.

From Magnitudes Proportional Separated are Proportional Compounded, $CA : AD = BA : AE$.

From Proportional Magnitudes are Proportional Alternately $CA : BA = AD : AE$.

From Ratio Equals its Multiples $AD : AE = n \cdot AD : n \cdot AE$, where $n$ is the number of times $AD$ is contained in $AC$.

From Equality of Ratios is Transitive, $AC : AB = n \cdot AD : n \cdot AE$.

So from Relative Sizes of Components of Ratios it follows that $AB = n \cdot AE$.

$\blacksquare$

Historical Note

This is Proposition 9 of Book VI of Euclid's The Elements.

This proposition is in fact only a particular case of Book VI Proposition 10.
The proof given here is that given by Robert Simson, with a refinement by Camerer, as Euclid's original was unnecessarily specific.