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 $a : b = c : d$.

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

 * Euclid-VI-12.png

From Parallel Transversal Theorem:
 * $DG : GE = DH : HF$

But:
 * $DG = A, GE = B, DH = C$

Hence the result.