We study how the supporting hyperplanes produced by the projection process can complement the method of alternating projections and its variants for the convex set intersection problem. Based on this idea, we propose an algorithm that, like Dykstra's algorithm, converges strongly in a Hilbert space to the intersection of finitely many convex sets. An analogue of the alternating projection algorithm can converge superlinearly with few added conditions, or quadratically under additional regularity. Under a conical condition, the convergence is finite. We present results of our numerical experiments.