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In particular, for three points in the plane (n = 2), the above matrix is square and the points are collinear if and only if its determinant is zero; since that 3 × 3 determinant is plus or minus twice the area of a triangle with those three points as vertices, this is equivalent to the statement that the three points are collinear if and only ...
Equivalently, this is the smallest set that could be produced by a greedy algorithm that tries to solve the no-three-in-line problem by placing points one at a time until it gets stuck. [3] If only axis-parallel and diagonal lines are considered, then every such set has at least n − 1 {\displaystyle n-1} points. [ 18 ]
A triangle is given by three non-collinear points (called vertices) and their three segments AB, BC, and CA. If three points A, B, and C are non-collinear, then a plane ABC is the set of all points collinear with pairs of points on one or two of the sides of triangle ABC.
Four points do not determine a conic, but rather a pencil, the 1-dimensional linear system of conics which all pass through the four points (formally, have the four points as base locus). Similarly, three points determine a 2-dimensional linear system (net), two points determine a 3-dimensional linear system (web), one point determines a 4 ...
In particular, for three points in the plane (n = 2), the above matrix is square and the points are collinear if and only if its determinant is zero. Equivalently for three points in a plane, the points are collinear if and only if the slope between one pair of points equals the slope between any other pair of points (in which case the slope ...
Similarly, three generic points in the plane are not collinear; if three points are collinear (even stronger, if two coincide), this is a degenerate case. This notion is important in mathematics and its applications, because degenerate cases may require an exceptional treatment; for example, when stating general theorems or giving precise ...
Given a plane in space, there exists at least one point in space that is not in the plane. Flat plane assumption. If two points lie in a plane, the line containing them lies in the plane. Unique plane assumption. Through three non-collinear points, there is exactly one plane. Intersecting planes assumption.
If the three points are collinear, R can be informally considered to be +∞, and it makes rigorous sense to define c(x, y, z) = 0. If any of the points x, y and z are coincident, again define c(x, y, z) = 0. Using the well-known formula relating the side lengths of a triangle to its area, it follows that