Construction of Fourth Proportional Straight Line
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Theorem
Given three straight lines of lengths $a, b, c$, it is possible to construct a fourth straight line of length $d$ such that $a : b = c : d$.
In the words of Euclid:
- To three given straight lines to find a fourth proportional.
(The Elements: Book $\text{VI}$: 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 Transversal Theorem:
- $DG : GE = DH : HF$
But:
- $DG = A, GE = B, DH = C$
Hence the result.
$\blacksquare$
Historical Note
This proof is Proposition $12$ of Book $\text{VI}$ of Euclid's The Elements.
Sources
- 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 2 (2nd ed.) ... (previous) ... (next): Book $\text{VI}$. Propositions