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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 ...
In geometry, given a triangle ABC and a point P on its circumcircle, the three closest points to P on lines AB, AC, and BC are collinear. [1] The line through these points is the Simson line of P, named for Robert Simson. [2] The concept was first published, however, by William Wallace in 1799, [3] and is sometimes called the Wallace line. [4]
By extension, k points in a plane are collinear if and only if any (k–1) pairs of points have the same pairwise slopes. In Euclidean geometry, the Euclidean distance d(a,b) between two points a and b may be used to express the collinearity between three points by: [3] [4]
They proved that the maximum number of points in the grid with no three points collinear is (). Similarly to Erdős's 2D construction, this can be accomplished by using points ( x , y , x 2 + y 2 {\displaystyle (x,y,x^{2}+y^{2}} mod p ) {\displaystyle p)} , where p {\displaystyle p} is a prime congruent to 3 mod 4 . [ 20 ]
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
Points of the line X = Y have coordinates which may be written as (1,1,c). Three points, one from each of these lines, are collinear if and only if a = b + c. By selecting all the points on these lines where a, b and c are the field elements with absolute trace 0, the condition in the definition of a projective triad is satisfied.
Monge's theorem states that the three such points given by the three pairs of circles always lie in a straight line. In the case of two of the circles being of equal size, the two external tangent lines are parallel. In this case Monge's theorem asserts that the other two intersection points must lie on a line parallel to those two external ...
In conclusion, =, and so the three points , and (in this order) are collinear. In Dörrie's book, [7] the Euler line and the problem of Sylvester are put together into a single proof. However, most of the proofs of the problem of Sylvester rely on the fundamental properties of free vectors, independently of the Euler line.