Euclid’s Elements — Book XIII, Proposition 4

Golden Ratio Geometry: A Triple-Square Identity

In Book XIII of Euclid’s Elements, Proposition 4 presents a beautiful geometric relationship that emerges when a straight line is divided according to the extreme and mean ratio — what we now know as the golden ratio.

This result connects the squares on the whole line, the lesser segment, and the greater segment in an elegant proportional identity.

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Proposition Statement

If a straight line is cut in extreme and mean ratio, then the sum of the squares on the whole and on the lesser segment equals triple the square on 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 being the greater segment.

• We want to prove:

  $$A{B}^2+B{C}^2=3\cdot A{C}^2$$

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Key Result

Euclid proves the following identity:

$$A{B}^2+B{C}^2=3\cdot A{C}^2$$

In words:

The sum of the squares on the whole line and the lesser segment equals three times the square on the greater segment.

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Sketch of the Proof

Euclid’s argument relies on geometric algebra — translating relationships between areas into proportions between line segments:

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

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

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

2. Equality of Areas
   Using squares and rectangles constructed on the relevant segments, Euclid shows that:

   $$AK=HG,$$

   where AK represents the rectangle $AB \cdot BC$ and HG represents the square on AC.

3. Combining Areas
   By rearranging gnomons (L-shaped areas) and squares within the main figure, Euclid derives the equality:

   $$A{B}^2+B{C}^2=3\cdot A{C}^2\mathrm{.}$$

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Golden Ratio Connection

This proposition beautifully encodes the golden ratio $\phi$ into a Pythagorean-like identity.
Since:

$$\frac{AB}{AC}=\varphi ,$$

we can rewrite the relationship as:

$$A{B}^2+B{C}^2=3\cdot A{C}^2,$$

which connects the segments in a ratio involving $\phi$ and ultimately relates to pentagon geometry.

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Summary

If a straight line is divided in the golden ratio,
then the square on the whole line plus the square on the lesser segment equals three times the square on the greater segment.

$$\boxed{{\displaystyle A{B}^2+B{C}^2=3\cdot A{C}^{\mathrm{2}}}}$$

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Why This Matters

This proposition reveals another deep connection between the golden ratio and geometric constructions.

It forms part of the foundation for the study of regular polygons — especially the pentagon, decagon, and icosahedron — which dominate Book XIII of Euclid’s Elements.