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Corollary: 6.20: Similar Polygons are Composed of Similar Triangles

(related to Proposition: 6.20: Similar Polygons are Composed of Similar Triangles)

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

And, in the same manner, it can also be shown for [similar quadrilaterals that they are in the squared ratio of (their) corresponding sides. And it was also shown for triangles. Hence, in general, similar rectilinear figures are also to one another in the squared ratio of (their) corresponding sides. (Which is) the very thing it was required to show.

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

Table of Contents

Proofs: 1

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

Github:
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non-Github:
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References

Adapted from (subject to copyright, with kind permission)

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

Footnotes

  1. Euclid formulates this corollary as a generalization of Prop. 6.20 with a triangularization argument and a reference to Prop 6.19 (editor's note). ↩