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is of rank 2 or less, the points are collinear. 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 ...
The no-three-in-line drawing of a complete graph is a special case of this result with =. [12] The no-three-in-line problem also has applications to another problem in discrete geometry, the Heilbronn triangle problem. In this problem, one must place points, anywhere in a unit square, not restricted to a grid. The goal of the placement is to ...
The three types of degenerate triangles, all of which contain zero area. A degenerate triangle is a "flat" triangle in the sense that it is contained in a line segment. It has thus collinear vertices [3] and zero area. If the three vertices are pairwise distinct, it has two 0° angles and one 180° angle.
In the case of three non-collinear points in the plane, the triangle with these points as its vertices has a unique Steiner inellipse that is tangent to the triangle's sides at their midpoints. The major axis of this ellipse falls on the orthogonal regression line for the three vertices. [7]
The Simson line LN (red) of the triangle ABC with respect to point P on the circumcircle. 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]
This maximum is attained for simple arrangements, those in which each two lines cross at a vertex that is disjoint from all the other lines. The number of vertices is smaller when some lines are parallel, or when some vertices are crossed by more than two lines. [4] An arrangement can be rotated, if necessary, to avoid axis-parallel lines.
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 ...
The nine vertices of three colored triangles (red, blue, and yellow) regrouped into three rainbow triangles (black edges) Rota's basis conjecture has a simple formulation for points in the Euclidean plane: it states that, given three triangles with distinct vertices, with each triangle colored with one of three colors, it must be possible to regroup the nine triangle vertices into three ...