Proposition XIII.8 — If in an equilateral and equiangular pentagon straight lines…

Proof

Construct the pentagon inscribed in a circle (IV.11). Two diagonals form an isosceles triangle with vertex angle

3 6^{\circ}

(I.32 / IV.10); by similarity (VI.4) the diagonal-segment ratio matches the extreme-and-mean ratio.

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Depends on (4)

IV.10Proposition IV.10To construct an isosceles triangle having each of the angles at the base double of the remaining one.
IV.11Proposition IV.11In a given circle to inscribe an equilateral and equiangular pentagon.
VI.4Proposition VI.4In equiangular triangles the sides about the equal angles are proportional, and those are corresponding sides which…
XIII.1Definition XIII.1A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment,…

Required by (dependents) (1)

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…

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