Proof
Construct the 72–72–36 isosceles triangle by IV.10. Inscribe
in the given circle a triangle equiangular with (IV.2).
Bisect the base-angles of by IV.10's construction propagated
into the circle, yielding two more division points , . The
five arcs are equal (III.26), so the five chords , , ,
, are equal (III.29), and the inscribed angles standing on
equal arcs are equal (III.27): the pentagon is regular.
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Full neighborhood
Depends on (6)
- I.9Proposition I.9To bisect a given rectilineal angle.
- IV.2Proposition IV.2In a given circle to inscribe a triangle equiangular with a given triangle.
- IV.10Proposition IV.10To construct an isosceles triangle having each of the angles at the base double of the remaining one.
- III.26Proposition III.26In equal circles equal angles stand on equal circumferences, whether they stand at the centres or at the circumferences.
- III.27Proposition III.27In equal circles angles standing on equal circumferences are equal to one another, whether they stand at the centres or…
- III.29Proposition III.29In equal circles equal circumferences are subtended by equal straight lines.
Required by (dependents) (6)
- IV.12Proposition IV.12About a given circle to circumscribe an equilateral and equiangular pentagon.
- IV.16Proposition IV.16In a given circle to inscribe a fifteen-angled figure which shall be both equilateral and equiangular.
- XIII.8Proposition XIII.8If in an equilateral and equiangular pentagon straight lines subtend two adjacent angles, they cut one another in…
- 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…
- XIII.16Proposition XIII.16To construct an icosahedron and comprehend it in a sphere, as in the case of the aforesaid figures; and to prove that…
- XIII.17Proposition XIII.17To construct a dodecahedron and comprehend it in a sphere, like the aforesaid figures; and to prove that the side of…
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