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.

Let $AB$, $CD$, $EF$, and $GH$ be four proportional straight lines, (such that) as $AB$ (is) to $CD$, so $EF$ (is) to $GH$.
And let the similar, and similarly laid out, rectilinear figures $KAB$ and $LCD$ have been described on $AB$ and $CD$ (respectively), and the similar, and similarly laid out, rectilinear figures $MF$ and $NH$ on $EF$ and $GH$ (respectively).
I say that as $KAB$ is to $LCD$, so $MF$ (is) to $NH$ (and conversely).

fig22e

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

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