Let M N be a chord of a circle, and let S be its midpoint. Now let A, B, C, D be points on that circle such that A C and B D both contain S, and A and B are on the same side of M N. Let d_{A}, d_{B}, d_{C}, d_{D} be the distances from A, B, C, D respectively to M N. Prove that \frac{1}{d_{A}}+\frac{1}{d_{D}}=\frac{1}{d_{B}}+\frac{1}{d_{C}}.