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.
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.