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Proposition: Prop. 13.13: Construction of Regular Tetrahedron within Given Sphere

(Proposition 13 from Book 13 of Euclid's “Elements”)

To construct a (regular) pyramid (i.e., a tetrahedron), and to enclose (it) in a given sphere, and to show that the square on the diameter of the sphere is one and a half times the (square) on the side of the pyramid.

• Thus, [it is required to construct] a pyramid, whose base is triangle $EFG$, and apex the point $K$, from four "equilateral triangles":https://www.bookofproofs.org/branches/equilateral-triangle-isosceles-triangle-scalene-triangle/...
• So, I say that the square on the diameter of the sphere is one and a half times the (square) on the side of the pyramid.

Modern Formulation

(not yet contributed)

Notes

If the radius of the sphere is unity then the side of the pyramid (i.e., tetrahedron) is \[\sqrt{\frac 83}.\]

Table of Contents

Proofs: 1

1. Lemma: Lem. 13.13: Construction of Regular Tetrahedron within Given Sphere

Mentioned in:

Proofs: 1

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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