(related to Proposition: 6.22: Similar Figures on Proportional Straight Lines)

- 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 $NH$.
- For let a third (straight line) $O$ have been taken (which is) proportional to $AB$ and $CD$, and a third (straight line) $P$ proportional to $EF$ and $GH$ [Prop. 6.11].
- And since as $AB$ is to $CD$, so $EF$ (is) to $GH$, and as $CD$ (is) to $O$, so $GH$ (is) to $P$, thus, via equality, as $AB$ is to $O$, so $EF$ (is) to $P$ [Prop. 5.22].
- But, as $AB$ (is) to $O$, so [also] $KAB$ (is) to $LCD$, and as $EF$ (is) to $P$, so $MF$ (is) to $NH$ [Prop. 5.19 corr.] 2.
- And, thus, as $KAB$ (is) to $LCD$, so $MF$ (is) to $NH$.

- And so let $KAB$ be to $LCD$, as $MF$ (is) to $NH$.
- I say also that as $AB$ is to $CD$, so $EF$ (is) to $GH$.
- For if as $AB$ is to $CD$, so $EF$ (is) not to $GH$, let $AB$ be to $CD$, as $EF$ (is) to $QR$ [Prop. 6.12].
- And let the rectilinear figure $SR$, similar, and similarly laid down, to either of $MF$ or $NH$, have been described on $QR$ [Prop. 6.18], [Prop. 6.21].
- Therefore, since as $AB$ is to $CD$, so $EF$ (is) to $QR$, and 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 $SR$ on $EF$ and $QR$ (resespectively), thus as $KAB$ is to $LCD$, so $MF$ (is) to $SR$ (see above).
- And it was also assumed that as $KAB$ (is) to $LCD$, so $MF$ (is) to $NH$.
- Thus, also, as $MF$ (is) to $SR$, so $MF$ (is) to $NH$ [Prop. 5.11].
- Thus, $MF$ has the same ratio to each of $NH$ and $SR$.
- Thus, $NH$ is equal to $SR$ [Prop. 5.9].
- And it is also similar, and similarly laid out, to it.
- Thus, $GH$ (is) equal to $QR$.
^{1} - And since $AB$ is to $CD$, as $EF$ (is) to $QR$, and $QR$ (is) equal to $GH$, thus as $AB$ is to $CD$, so $EF$ (is) to $GH$.

- Thus, 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.
- (Which is) the very thing it was required to show.

∎

**Fitzpatrick, Richard**: Euclid's "Elements of Geometry"