Proposition XIII.9 — If the side of the hexagon and that of the decagon inscribed…

Proof

The hexagon side equals the radius (IV.15); the decagon side satisfies the 36-72-72 triangle relations (IV.10); their sum is in the golden ratio to the hexagon side.

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Depends on (3)

IV.10Proposition IV.10To construct an isosceles triangle having each of the angles at the base double of the remaining one.
IV.15Proposition IV.15In a given circle to inscribe an equilateral and equiangular hexagon.
XIII.8Proposition XIII.8If in an equilateral and equiangular pentagon straight lines subtend two adjacent angles, they cut one another in…

Required by (dependents) (1)

XIII.10Proposition XIII.10If an equilateral pentagon be inscribed in a circle, the square on the side of the pentagon is equal to the squares on…

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