Ratio of Lengths of Arms of Pentagram

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Theorem

Consider a pentagram.

PentagramArmLengths.png

Let $AC$ be the length of one of the lines which span the pentagram and define it.

Let $B$ be one of the points where $AC$ intersects one of the other such lines such that $AB > AC$.

Then:

$\dfrac {AC} {AB} = \phi$

where $\phi$ denotes the golden mean.


Proof

Follows directly from Straight Lines Subtending Two Consecutive Angles in Regular Pentagon cut in Extreme and Mean Ratio.

$\blacksquare$


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Historical Note

The Ratio of Lengths of Arms of Pentagram was known to the Pythagoreans.


Sources

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