1. ## constructible angle

The measure of a given angle is
180o

n
, where n is a positive integer
not divisible by 3. Prove that this angle can be trisected by Eucliden means

(straightedge and compass).

2. Originally Posted by anncar
The measure of a given angle is
180o
n
, where n is a positive integer
not divisible by 3. Prove that this angle can be trisected by Eucliden means

(straightedge and compass).
3. It is possible to construct a regular pentagon whose angles are 72 and an equilateral triangles with angles 60. So it is possible to construct 72 - 60 = 12degrees. Bisect this angle with compass to get that 6 is constructible. Bisect again to show that 3 is constructible. However, 1 is not constructible because 3 cannot be trisected*. Also 2 cannot be constructed for that will imply that 1 (after bisection) can be constructed a contradiction. Thus, the smallest contructible angle is 3 degrees. Then any multiple of this can be constructed. Since $180/n$ does not contain a factor of 3 it means after this division this integer is a multiple of three. Thus, it can be constructed. Q.E.D.
*)Because constructible numbers are algebraic over $\mathbb{Q}$ of degree power of two. Note that $4\cos^3 x - 3\cos x - \cos 3x = 0$ showing that this polynomial is of degree 3 over the rationals.