Construction of Fourth Proportional Straight Line

Theorem

Given three straight lines of lengths $a, b, c$, it is possible to construct a fourth straight line of length $d$ such that $a : b = c : d$.

As Euclid defined it:

To three given straight lines to find a fourth proportional.

(The Elements: Book VI: Proposition $12$)

Construction

Let $A, B, C$ be the three given straight lines.

Let $DE, DF$ be set out containing any angle $EDF$.

Let $DG = A, GE = B, DH = C$.

Join $GH$ and construct $EF$ parallel to $GH$.

Then $HF$ is the required straight line such that $A : B = C : HF$.

Proof

From Parallel Line in Triangle Cuts Sides Proportionally we have $DG : GE = DH : HF$.

But $DG = A, GE = B, DH = C$.

Hence the result.

$\blacksquare$

Historical Note

This is Proposition 12 of Book VI of Euclid's The Elements.