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Proposition: 1.20: The Sum of the Lengths of Any Pair of Sides of a Triangle (Triangle Inequality)

(Proposition 20 from Book 1 of Euclid's “Elements”)

In any triangle, (the sum of) two sides taken together in any (possible way) is greater than the remaining (side).

• For let $ABC$ be a triangle.
• I say that in triangle $ABC$ (the sum of) two sides taken together in any (possible way) is greater than the remaining (side).

Modern Formulation

see triangle inequality.

Table of Contents

Proofs: 1

Mentioned in:

Proofs: 1 2 3 4 5 6 7 8
Propositions: 9
Sections: 10

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

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non-Github:

@Calahan

@Casey

@Fitzpatrick

References

Adapted from CC BY-SA 3.0 Sources:

1. Callahan, Daniel: "Euclid’s 'Elements' Redux" 2014

Adapted from (Public Domain)

1. Casey, John: "The First Six Books of the Elements of Euclid"

Adapted from (subject to copyright, with kind permission)

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