The Sylvester-Gallai Theorem
The deformation is the mathematical concept.
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 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 must be the ordinary line. This is the contradiction.
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.
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.
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