Proposition IV.15 — In a given circle to inscribe an equilateral and equiangular…

Proof

Let

G

be the centre and

A D

a diameter of the given circle. With centre

D

and radius

D G

describe a circle meeting the given circle at

C

and

E

(Postulate 3). Join

GC

GE

. The triangle

G D C

has

G D = D C = C G

(radii of equal circles), so it is equilateral, and

∠ D GC = 6 0^{\circ}

(I.32). Stepping this

6 0^{\circ}

chord around the circle six times produces the regular hexagon.

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Full neighborhood

Depends on (3)

I.32Proposition I.32In any triangle, if one of the sides be produced, the exterior angle is equal to the two interior and opposite angles,…
3Postulate 3To describe a circle with any centre and distance.
I.15Definition I.15A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among…

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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