Euclid’s Elements — Book XIII, Proposition 10

The Square on the Side of a Pentagon Equals the Sum of the Squares on the Sides of the Hexagon and the Decagon in the Same Circle

One of the most elegant results from Book XIII of Euclid’s Elements reveals a remarkable relationship between three regular polygons inscribed in the same circle: the pentagon, the hexagon, and the decagon.

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Proposition Statement

If an equilateral pentagon is inscribed in a circle, then the square on its side equals the sum of the squares on the sides of the hexagon and the decagon inscribed in the same circle.

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Geometric Setup

• Consider a circle ABCDE.

• Inside the circle, inscribe an equilateral pentagon ABCDE.

• Also inscribe:

  • A regular hexagon in the same circle.

  • A regular decagon in the same circle.

• Let the side of the pentagon be AB, the side of the hexagon be BF, and the side of the decagon be AK.

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Key Result

Euclid proves that:

$$A{B}^2=B{F}^2+A{K}^2$$

In other words:

The square on the side of the pentagon equals the sum of the squares on the sides of the hexagon and the decagon.

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Sketch of the Proof

The proof involves several classical geometric constructions:

1. Auxiliary Points and Symmetries

   • The center of the circle F is used to construct perpendiculars, auxiliary points, and proportional segments.

   • The symmetry of the circle allows relationships between the arcs and chords to be derived.

2. Properties of Inscribed Polygons

   • In a circle, the sides of regular polygons are related through their subtended central angles:

     • Hexagon side subtends an angle of 60°.

     • Decagon side subtends an angle of 36°.

     • Pentagon side subtends an angle of 72°.

   • Using these relationships, Euclid connects the chords geometrically.

3. Application of Similar Triangles
   Through a series of equiangular triangles, proportionality relationships are established between the segments:

   $$\frac{AB}{BF}=\frac{BF}{BN},\frac{BA}{AK}=\frac{AK}{AN}\mathrm{.}$$

   These proportionalities lead to equations involving squares on the sides.

4. Final Identity
   Combining the proportions and using properties of rectangles and squares, Euclid arrives at:

   $$A{B}^2=B{F}^2+A{K}^2\mathrm{.}$$

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Mathematical Significance

This result beautifully links three different regular polygons inscribed in the same circle. It highlights how Euclid’s geometry reveals harmonies between seemingly unrelated figures.

• Pentagon side ↔ related to the golden ratio.

• Hexagon side ↔ directly equal to the circle’s radius.

• Decagon side ↔ connected to divisions of the circle by 36° arcs.

Thus, the proposition reveals a Pythagorean-like relationship among the three polygons.

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Summary

If you inscribe an equilateral pentagon, a hexagon, and a decagon in the same circle,

the square on the side of the pentagon equals the sum of the squares on the sides of the hexagon and the decagon.

$$\boxed{{\displaystyle A{B}^2=B{F}^2+A{K}^{\mathrm{2}}}}$$