# Proposition: 6.20: Similar Polygons are Composed of Similar Triangles

### (Proposition 20 from Book 6 of Euclid's “Elements”)

Similar polygons can be divided into equal numbers of similar triangles corresponding (in proportion) to the wholes, and one polygon has to the (other) polygon a squared ratio with respect to (that) a corresponding side (has) to a corresponding side. * Let $ABCDE$ and $FGHKL$ be similar polygons, and let $AB$ correspond to $FG$. * I say that polygons $ABCDE$ and $FGHKL$ can be divided into equal numbers of similar triangles corresponding (in proportion) to the wholes, and (that) polygon $ABCDE$ has a squared ratio to polygon $FGHKL$ with respect to that $AB$ (has) to $FG$.

### Modern Formulation

The ratio of the areas of two similar rectilinear figures is proportional to the squared ratio of the lengths of corresponding sides.1

Proofs: 1 Corollaries: 1

Corollaries: 1
Proofs: 2 3 4

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### References

#### 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", https://proofwiki.org/wiki/Main_Page, 2016

#### Footnotes

1. Euclid formulates this proposition generally for all similar rectilinear figures but proves it for pentagonal figures only (editor's note).