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Proposition: Prop. 13.11: Side of Regular Pentagon inscribed in Circle with Rational Diameter is Minor
(Proposition 11 from Book 13 of Euclid's “Elements”)
If an equilateral pentagon is inscribed in a circle which has a rational diameter then the side of the pentagon is that irrational (straight line) called minor.
Modern Formulation
If the circle has unit radius then the side of the pentagon is \[\frac{\sqrt{102\,\sqrt{5}}}2.\]
However, this length can be written in the minor.
\[\sqrt{5}\sqrt{\left(1+\frac{\rho}{\sqrt{1+\rho^2}}\right)\frac 12}  \sqrt{\left(1\frac{\rho}{\sqrt{1+\rho^2}}\right)\frac 12}, \]
with $\rho=2$.
Table of Contents
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Proofs: 1
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References
Adapted from (subject to copyright, with kind permission)
 Fitzpatrick, Richard: Euclid's "Elements of Geometry"
Adapted from CC BYSA 3.0 Sources:
 Prime.mover and others: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016