Constructible Length with Compass and Straightedge

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