# Constructible Length with Compass and Straightedge

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## Theorem

Let $L$ be a line segment in a Eucldiean space.

Let the length of $L$ be $d$.

Let $L'$ be a line segment of length $d'$ constructed from $L$ using a compass and straightedge construction.

Then:

- $d' = q d$

where $q$ is an algebraic number whose degree is a power of $2$.

## Proof

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## Sources

- 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.18$: Algebraic and Transcendental Numbers. $e$ is Transcendental