  The Sylvester-Gallai Theorem, stated as a problem by J. J. Sylvester in 1893,
asserts that for any finite, noncollinear set of points on a plane, there
exists a line passing through exactly two points of the set. First, it is shown
that for the real plane, the theorem is constructively invalid. Then, a
well-known classical proof is examined from a constructive standpoint, locating
the nonconstructivities. Finally, a constructive version of the theorem is
established for the plane; this reveals the hidden constructive content of the
classical theorem. The constructive methods used are those proposed by Errett
Bishop.
