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Proposition: 1.48: The Converse of the Pythagorean Theorem

(Proposition 48 from Book 1 of Euclid's “Elements”)

If the square on one of the sides of a triangle is equal to the (sum of the) squares on the two remaining sides of the triangle then the angle contained by the two remaining sides of the triangle is a right angle.

• For let the square on one of the sides, $BC$, of triangle $ABC$ be equal to the (sum of the) squares on the sides $BA$ and $AC$.
• I say that angle $BAC$ is a right angle.

Modern Formulation

If the square on one side (\(\overline{BC}\)) of a triangle (\(\triangle{ABC}\)) equals the sum of the squares on the remaining sides (\(\overline{BA}\), \(\overline{AC}\)), then the angle (\(\angle{CAB}\)) opposite to that side is a right angle.

Table of Contents

Proofs: 1

Mentioned in:

Proofs: 1
Sections: 2

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

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@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"