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I have a problem concerning the necessary and sufficient condition for a regular n-gon to be constructible by ruler and compass.

$\bf My$ $\bf question:$ For a given positive integer $n$, how can we prove that a regular $n$-gon is constructible by ruler and compass if and only if the number $cos(2\pi/n)$ (that is, the central angle) is constructible?

Thanks very much!

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Suppose that your regular polygon has a vertice on the $x$-axis. Then the first vertice counted counter clock wise has coordinates $(\cos(\frac{2\pi}{n}),\sin(\frac{2\pi}{n}))$.

Hence if you know the construction of the polygon, by projecting the first vertice on the $x$ axis, which can be done with a ruler and a compass, you can get $\cos(\frac{2\pi}{n})$.

Conversely, if you have a construction of $\cos(\frac{2\pi}{n})$, by intersecting the unit circle with the vertical line passing at point $(\cos(\frac{2\pi}{n}), 0)$ you get the first vertice of the regular polygon. Therefore the angle which enables to build the full polygon.

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  • $\begingroup$ Thank you very much! This is a very interesting and clear answer! $\endgroup$ May 24 '15 at 6:46

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