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The Sylvester–Gallai theorem was posed as a problem by J. J. Sylvester (). Kelly () suggests that Sylvester may have been motivated by a related phenomenon in algebraic geometry, in which the inflection points of a cubic curve in the complex projective plane form a configuration of nine points and twelve lines (the Hesse configuration) in which each line determined by two of the points ...
Given two different points (x 1, y 1) and (x 2, y 2), there is exactly one line that passes through them. There are several ways to write a linear equation of this line. If x 1 ≠ x 2, the slope of the line is . Thus, a point-slope form is [3]
Let (m, n) be the point of intersection of the line ax + by + c = 0 and the line perpendicular to it which passes through the point (x 0, y 0). The line through these two points is perpendicular to the original line, so =.
In more general Euclidean space, R n (and analogously in every other affine space), the line L passing through two different points a and b is the subset = {() +}. The direction of the line is from a reference point a ( t = 0) to another point b ( t = 1), or in other words, in the direction of the vector b − a .
For example, if K is a set of 50 points arranged on a circle in the Euclidean plane, a line joining two of them would be a 2-secant (or bisecant) and a line passing through only one of them would be a 1-secant (or unisecant). A unisecant in this example need not be a tangent line to the circle.
If a chord were to be extended infinitely on both directions into a line, the object is a secant line. The perpendicular line passing through the chord's midpoint is called sagitta (Latin for "arrow"). More generally, a chord is a line segment joining two points on any curve, for instance, on an ellipse.
Through any point other than a base point, there passes a single conic of the pencil. This concept generalizes a pencil of circles. In a projective plane defined over an algebraically closed field any two conics meet in four points (counted with multiplicity) and so, determine the pencil of conics based on these four points.
A subset of P(V) is a projective subspace if and only if, given any two different points, it contains the whole projective line passing through these points. In synthetic geometry , where projective lines are primitive objects, the first property is an axiom, and the second one is the definition of a projective subspace.