Euclid’s Elements — Book XIII, Proposition 9

The Sum of the Sides of the Hexagon and Decagon Divides a Line in Extreme and Mean Ratio

Book XIII of Euclid’s Elements is dedicated to the geometry of regular polygons and the five Platonic solids. In Proposition 9, Euclid reveals a beautiful relationship between the hexagon and decagon inscribed in the same circle.

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Proposition Statement

If the side of the hexagon and that of the decagon inscribed in the same circle are added together, then the entire straight line is cut in extreme and mean ratio, and the greater segment is the side of the hexagon.

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Geometric Setup

• Consider a circle ABC.

• Inside the circle:

  • BC = side of the decagon.

  • CD = side of the hexagon.

• Arrange them so that BC and CD lie on the same straight line.

• Let BD be the sum of the two:

  $$BD=BC+CD$$

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Key Result

Euclid proves that:

$$\frac{BD}{DC}=\frac{DC}{C\mathrm{B}}$$

This means that the straight line BD is divided in extreme and mean ratio — what we know today as the golden ratio. Moreover, the greater segment is the side of the hexagon (CD).

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Sketch of the Proof

The proof relies on relationships between central angles and chords in the circle:

1. Arcs and Angles

   • Since BC is the side of a decagon, the arc subtended by BC is $\dfrac{1}{10}$ of the circle.

   • Using proportionality between central angles and arcs, Euclid establishes precise angular relationships.

2. Equal Chords and Angles

   • Because CD equals the side of the hexagon, EC = CD.

   • From symmetry, several triangles are shown to be equiangular.

3. Proportional Triangles
   Using similar triangles, Euclid derives:

   $$BD:DC=DC:CB$$

   which is the defining property of the extreme and mean ratio.

4. Identification of Segments
   Since BD > DC > CB, Euclid concludes that:

   • BD = whole line.

   • DC = greater segment = side of the hexagon.

   • BC = smaller segment = side of the decagon.

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Mathematical Significance

This proposition links the geometry of regular polygons to the golden ratio:

$$\varphi =\frac{BD}{DC}=\frac{DC}{C\mathrm{B}}$$

• The golden ratio arises naturally when combining the sides of the hexagon and decagon.

• This result is central in understanding the geometry behind the pentagon, decagon, and icosahedron, which are explored further in Book XIII.

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Summary

When the side of the hexagon and the side of the decagon inscribed in the same circle are added, the resulting straight line is divided in extreme and mean ratio.
The greater segment equals the side of the hexagon.

$$\boxed{{\displaystyle \frac{BD}{DC}=\frac{DC}{CB}}}$$