Euclid’s Elements — Book XIII, Proposition 5

Extending a Golden Section Creates Another Golden Section

In Book XIII of Euclid’s Elements, Proposition 5 reveals a fascinating recursive property of the extreme and mean ratio — what we now call the golden ratio.

If a straight line is divided in the golden ratio and we extend it by adding a segment equal to the greater part, the new, longer line is also divided in the golden ratio.

This result beautifully shows how the golden ratio repeats itself within geometric constructions.

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Proposition Statement

If a straight line is cut in extreme and mean ratio, and a straight line equal to the greater segment is added to it, then the whole line is again cut in extreme and mean ratio, and the original straight line is 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.

• Extend the line by adding AD, where:

  $$AD=AC$$

• Let the new longer line be DB.

• We want to prove:

  • The line DB is cut in extreme and mean ratio at A.

  • The original AB is the greater segment of DB.

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Key Result

Euclid proves the following relationship:

$$\frac{DB}{BA}=\frac{BA}{A\mathrm{D}}$$

Since $AD = AC$, this means the new, longer line DB is divided in the golden ratio at A.

Moreover, the original line AB is the greater segment of DB.

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Sketch of the Proof

The proof relies on geometric algebra and properties of the golden ratio:

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

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

   Using constructed squares and rectangles, Euclid translates this property into an equality of areas.

2. Adding the Equal Segment
   With AD = AC, new relationships between the constructed rectangles are formed, leading to:

   $$DB\cdot DA=A{B}^2$$

3. Extreme and Mean Ratio
   From the equality above, Euclid derives the proportionality:

   $$\frac{DB}{BA}=\frac{BA}{A\mathrm{D}}$$

   proving that DB is divided in extreme and mean ratio at A.

4. Comparison of Segments
   Since DB > AB > AD, the greater segment of the division is the original AB.

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Golden Ratio Connection

This proposition beautifully captures the self-replicating property of the golden ratio $\phi$:

$${\varphi }^2=\varphi +1$$

If we set:

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

then extending the line by AC naturally creates a larger segment that is still divided in the same ratio.

This recursive property is one of the reasons the golden ratio appears so frequently in geometry, nature, and art.

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Summary

If a straight line AB is divided in the golden ratio and we extend it by adding a segment AD equal to the greater part,
then the new longer line DB is also divided in the golden ratio,
and the original line AB becomes the greater segment.

$$\boxed{\dfrac{DB}{BA} = \dfrac{BA}{AD}}$$

Why This Matters

This proposition shows that the golden ratio naturally repeats itself when extended.

It explains why the golden ratio arises in pentagonal geometry and Fibonacci sequences, where each step builds upon the previous one while preserving proportional harmony.