The Sylvester-Gallai Theorem is that a finite set of points in the plane have the proper line through any two of them, and it passes through a third point of the set. Moreover, it must be on the same line.
P is the the noncollinear finite set. S(P) is the set of connecting lines in P.
p∈P, s∈S(P)
p and s are perpendicular, so they are not the ordinary line. (s*,p*) is the smallest distance, and s* is in the ordinary line.
l∩P={p}
l must be the ordinary line. This is the contradiction.
l∩S(P)
There is the cyclically intersection points (p,x1).
x1,・・・xk is on l. Moreover it crosses S. This must be ordinary.
You also see the extra dimensions.
i lines are determined by P.
