Comment by lopotubez
19 hours ago
So the solutions to the n'th roots of unity of the form z^n = 1 are n-gons on the unit circle. This means for an arbitrary w in the complex field z^n = w gives solutions on a circle with radius n'th root of modulus w and are 2*pi/n apart so it is also an n-gon.
Pretty fun stuff to plot around using Geogebra to give a geometric intuition how the complex numbers behave.
I built these two interactive visualizations to gain some intuition for this. [1][2]
1: https://static.laszlokorte.de/complex-roots/ 2: https://static.laszlokorte.de/complex-plane/
Complex numbers are awesome for bashing through (some) IMO geometry problems. Points P, Q, R are collinear exactly when (p-q)/(p-r) is real, rotation is multiplication by exp(iϕ), etc.
Nice but I prefer the much powerful version of 3 points being collinear precisely when ((p-q)^2 + (q-r)^2 - (p-r)^2)^2 = 4(p-q)^2(q-r)^2 which works for any field(!) with non-characteristics 2.
If they are non-collinear/degenerate then the difference becomes 16 * area^2 of the triangle PQR, in which case one can easily derive Heron's formula.