Euclid’s Elements — Book XIII, Proposition 3

Golden Ratio Geometry: A Fivefold Square Relationship

In Book XIII of Euclid’s Elements, we continue exploring the fascinating properties of the extreme and mean ratio — what we now call the golden ratio.

Proposition 3 demonstrates a remarkable geometric identity: when a straight line is divided according to the golden ratio, a special relationship emerges between the greater and lesser segments and their halves.

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Proposition Statement

If a straight line is cut in extreme and mean ratio, then the square on the sum of the lesser segment and half of the greater segment equals five times the square on the half of the greater segment.

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Geometric Setup

• Take a straight line AB.

• Cut it in extreme and mean ratio at point C so that:

  $$\frac{AB}{AC}=\frac{AC}{C\mathrm{B}}$$

  with AC as the greater segment.

• Bisect AC at D, so:

  $$AD=DC=\frac{1}{2}AC$$

• The goal is to prove:

  $$B{D}^2=5\cdot D{C}^2$$

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Key Result

Euclid proves the identity:

$$B{D}^2=5\cdot D{C}^2$$

In words:

The square on the sum of the lesser segment (CB) and half the greater segment (DC) is exactly five times the square on half the greater segment.

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Sketch of the Proof

The proof uses geometric algebra — reasoning about areas of squares and rectangles rather than symbolic equations:

1. Golden Ratio Property
   Since AB is divided in extreme and mean ratio at C:

   $$AB\cdot BC=A{C}^2$$

   This relationship connects the whole line, the greater segment, and the lesser segment.

2. Halving the Greater Segment
   With AC bisected at D, we have:

   $$AC=2\cdot DC⟹A{C}^2=4\cdot D{C}^2$$

   Thus, key relationships involving the half-segment are established.

3. Equality of Areas
   Euclid constructs squares and rectangles on the relevant segments and shows that a certain gnomon (L-shaped area) equals four times the square on DC.

4. Final Equality
   Adding the appropriate rectangles and squares together, Euclid arrives at:

   $$B{D}^2=5\cdot D{C}^2$$

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Golden Ratio Connection

This proposition provides another beautiful identity arising from the golden ratio:

$$\boxed{{\displaystyle BD=\sqrt{5}\cdot D\mathrm{C}}}$$

Thus, the relationship between BD and DC directly encodes the appearance of $\sqrt{5}$, a constant deeply tied to pentagons, decagons, and icosahedra.

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Summary

If a straight line is divided in the golden ratio,
then the square on the sum of the lesser segment and half the greater segment equals five times the square on half the greater segment.

$$\boxed{{\displaystyle B{D}^2=5\cdot D{C}^{\mathrm{2}}}}$$

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Why This Matters

This proposition strengthens the connection between Euclidean geometry and the golden ratio.

It forms a crucial foundation for the results that follow in Book XIII, especially those involving the geometry of the pentagon and the regular icosahedron.