Suppose n runners, n >= 3, start at the same time and place on a circular track of circumference 1, and proceed to run counterclockwise along the track (forever). Assume the speeds v_1,v_2, ..., v_n of the runners, expressed in revolutions per unit time, are positive real numbers such that v_1 < v_2 < ... < v_n.
There is an instant of time where the locations of the n runners are the vertices of a regular n-gon iff each of the n fractions
(v_i - v_1)/(v_2 - v_1)
for i = 1,2,...,n is a rational number, and moreover, when reduced to lowest terms the n numerators yield all possible distinct residues mod n.
Speeds 1,2,4 cannot yield an equilateral triangle since, of the fractions