The regular 9-gon is not constructible by straightedge and compass

Prove that a regular 9-gon cannot be constructed by straightedge and compass.

Suppose to the contrary that we can construct a regular 9-gon. The internal angles of a 9-gon measure 140 degrees, so in particular a 140 degree angle is constructible. Given an angle, it is straightforward to construct its complement (simply extend one of the incident rays). So a 180-140 = 40 degree angle is also constructible, and bisecting, a 20 degree angle is constructible. However, as D&F argue on page 534, this is not possible. So a regular 9-gon cannot be constructed.

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  • John  On December 6, 2012 at 8:11 am

    You can’t simply refer to some random book, at least include an ISBN number, such that the book can be found. “D&F”, that is just some acronym, it could mean design and fabrication for all anyone cares ( Without a way to find said book, this is no proof.🙂

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