Proposition: 6.22: Similar Figures on Proportional Straight Lines

(Proposition 22 from Book 6 of Euclid's “Elements”)

If four straight lines are proportional then similar, and similarly described, rectilinear figures (drawn) on them will also be proportional. And if similar, and similarly described, rectilinear figures (drawn) on them are proportional then the straight lines themselves will also be proportional.


Modern Formulation

Let four similar rectilinear figures, $\mathcal A,\mathcal D,\mathcal C,\mathcal D$ with four correspondings sides be given: the first side in figure $\mathcal A$ of the length $a:=|\overline{AB}|$, the second side in figure $\mathcal B$ of the length $b:=|\overline{CD}|,$ the third side in figure $\mathcal C$ of the length $c:=|\overline{EF}|$ and the fourth side in figure $\mathcal D$ of the length $d:=|\overline{GH}|.$ Then these sides are proportional if and only if the areas of the figures are proportional, formally

$$\frac {a}{b}=\frac {c}{d}\Longleftrightarrow \frac {\operatorname{area}\mathcal A}{\operatorname{area}\mathcal B}=\frac {\operatorname{area}\mathcal C}{\operatorname{area}\mathcal D}.$$

Proofs: 1

Proofs: 1 2 3 4 5

Thank you to the contributors under CC BY-SA 4.0!



Adapted from (subject to copyright, with kind permission)

  1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"

Adapted from CC BY-SA 3.0 Sources:

  1. Prime.mover and others: "Pr∞fWiki",, 2016