Sylvester's line problem, known as the Sylvester-Gallai theorem in proved form, states that it is not possible to arrange a finite number of points so that a line
through every two of them passes through a third unless they are all on a single
line. This problem was proposed by Sylvester (1893), who
asked readers to "Prove that it is not possible to arrange any finite number
of real points so that a right line through every two of them shall pass through
a third, unless they all lie in the same right line."

Woodall (1893) published a four-line "solution," but an editorial comment following his result pointed out two holes in the argument and sketched another line
of enquiry, which is characterized as "equally incomplete, but may be worth
notice." However, no correct proof was published at the time (Croft et al.
1991, p. 159), but the problem was revived by Erdős (1943) and correctly
solved by Grünwald (1944). Coxeter (1948, 1969) transformed the problem into
an elementary form, and a very short proof using the notion of Euclidean distance
was given by Kelly (Coxeter 1948, 1969; Chvátal 2004). The theorem also follows
using projective duality from a result of Melchior (1940) proved by a simple application
of Euler's polyhedral formula (Chvátal
2004).

Additional information on the theorem can be found in Borwein and Moser (1990), Erdős and Purdy (1991), Pach and Agarwal (1995), and Chvátal (2003).

In September 2003, X. Chen proved a conjecture of Chvátal that, with a certain definition of a line, the Sylvester-Gallai theorem extends to arbitrary
finite metric spaces.

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H. S. M. §4.7 and 12.3 in Introduction
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