Several authors have considered the problem of finding rational points (x_i, y_i), i = 1, 2,..., n on various curves f(x, y) = 0, including conics, elliptic curves and hyperelliptic curves, such that the x-coordinates x_i, i = 1, 2,..., n are in arithmetic progression. In this paper we find infinitely many sets of three points, in parametric terms, on the unit circle x² + y² = 1 such that the x-coordinates of the three points are in arithmetic progression. It is an open problem whether there exist four rational points on the unit circle such that their x-coordinates are in arithmetic progression.