Proposition IV.16 — In a given circle to inscribe a fifteen-angled figure which …

Proof

Inscribe a regular pentagon (IV.11) and a regular equilateral triangle (IV.2) in the circle, sharing a common vertex

A

. The arc from

A

to the next pentagon-vertex is

\frac{1}{5}

of the circle; the arc from

A

to the next triangle-vertex is

\frac{1}{3}

. The difference is

\frac{1}{3} - \frac{1}{5} = \frac{2}{15}

of the circle. Bisect that arc (III.30); each half is

\frac{1}{15}

of the circle, and stepping that chord fifteen times around gives the regular pentadecagon.

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Full neighborhood

Depends on (3)

IV.2Proposition IV.2In a given circle to inscribe a triangle equiangular with a given triangle.
IV.11Proposition IV.11In a given circle to inscribe an equilateral and equiangular pentagon.
III.30Proposition III.30To bisect a given arc.

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