Construction of Fourth Proportional Straight Line

From ProofWiki
Jump to navigation Jump to search


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

In the words of Euclid:

To three given straight lines to find a fourth proportional.

(The Elements: Book $\text{VI}$: Proposition $12$)


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 $A : B = C : HF$.



From Parallel Transversal Theorem:

$DG : GE = DH : HF$


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

Hence the result.


Historical Note

This proof is Proposition $12$ of Book $\text{VI}$ of Euclid's The Elements.