Hacker News Reader

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.

Reply on news.ycombinator.com