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Proposition: 6.01: Areas of Triangles and Parallelograms Proportional to Base

(Proposition 1 from Book 6 of Euclid's “Elements”)1

Triangles and parallelograms which are of the same height are to one another as their bases. * Let $ABC$ and $ACD$ be triangles, and $EC$ and $CF$ parallelograms, of the same height $AC$. * I say that as base $BC$ is to base $CD$, so triangle $ABC$ (is) to triangle $ACD$, and parallelogram $EC$ to parallelogram $CF$.

fig01e

Modern Formulation

The proportion of areas of triangles (parallelograms) of the same height is equal to the proportion of the lengths of their bases.

Table of Contents

Proofs: 1

Mentioned in:

Proofs: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56

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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. As is easily demonstrated, this proposition holds even when the triangles, or parallelograms, do not share a common side, and/or are not right-angled (translator's note). ↩