Let $a_,\dots,a_n$ be positive real numbers and let $\omega$ be a primitive nth root of unity. If the sides of an equiangular polygon have lengths $a_,\dots,a_n$ (in counterclockwise order) then
$$1+\omega+\omega^2+\dots+\omega^{n-1}=0$$;
$$a_1 + a_2 \omega +a_3 \omega ^2+ \dots + a_n \omega^{n-1}=0$$.
Solat Idul Fitri di ‘s-Hertogenbosch
8 months ago
The first assertion follow from the definition of root of unity.
ReplyDeleteThe second follow from this :
Consider an equiangular $n$-gon which is it sides has lenght $a_1,a_2,\dots,a_n$, (counter clockwise order). Now, if the vector of it's sides we moved into origin, we will see that it's sides are $a_1, a_2 \omega , a_3 \omega^2, \dots, a_n \omega^{n-1}$, which is it's sum is 0.