Napoleon's Equilateral. This theorem is proposed by the famous Napoleon Bonaparte!!! Wow!! http://en.wikipedia.org/wiki/Napoleon's_theorem

suppose $O_1,O_2,O_3$ are centers of three $n$-gon face to $AB,BC,CA$ respectively.then use of complex numbers: $A=a,B=b,C=c,O_1=x,O_2=y,O_3=z, \omega$ is the third root of unity and $\alpha $ is the n,th root of unity,then: $\frac{x-a}{x-b}=\alpha,\frac{y-b}{y-c}=\alpha,\frac{z-c}{z-a}=\alpha$ now $O_1O_2O_3$ is equilateral so $x+y\omega+z\omega ^{2}=0$ now we have: $\alpha=\frac{a+b\omega+c\omega^{2}}{b+c\omega +a\omega ^{2}}=\omega$ so $n=3$.