Proposition XIII.10 — If an equilateral pentagon be inscribed in a circle, the squ…

Proof

This is the Pythagorean relation

p^{2} = h^{2} + d^{2}

in the inscribed polygons of a unit circle. Proven via I.47 applied to the right triangle formed by the centre, a pentagon-vertex, and a decagon-vertex.

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Depends on (3)

I.47Proposition I.47In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides…
IV.11Proposition IV.11In a given circle to inscribe an equilateral and equiangular pentagon.
XIII.9Proposition XIII.9If the side of the hexagon and that of the decagon inscribed in the same circle be added together, the whole straight…

Required by (dependents) (1)

XIII.11Proposition XIII.11If in a circle which has its diameter rational an equilateral pentagon be inscribed, the side of the pentagon is the…

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