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From any point on the circumcircle of a triangle, the nearest points on each of the three extended sides of the triangle are collinear in the Simson line of the point on the circumcircle. The lines connecting the feet of the altitudes intersect the opposite sides at collinear points. [3]: p.199
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] The converse is also true; if the three closest points to P on three lines are collinear, and no two of the lines are parallel, then P lies on ...
collinearity between points: three or more points which lie on the same line (called collinear points) continue to be collinear after the transformation. parallelism: two or more lines which are parallel, continue to be parallel after the transformation. convexity of sets: a convex set continues to be convex after the transformation.
Similarly, lines associated with a triangle are often constructed by proving that three symmetrically constructed points are collinear; here Menelaus' theorem gives a useful general criterion. [18] In this section, just a few of the most commonly encountered constructions are explained.
However, a set of four or more distinct points will, in general, not lie in a single plane. An example of coplanar points. Two lines in three-dimensional space are coplanar if there is a plane that includes them both. This occurs if the lines are parallel, or if they intersect each other. Two lines that are not coplanar are called skew lines.
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 ...
There are two types, points and lines, and one "incidence" relation between points and lines. The three axioms are: G1: Every line contains at least 3 points; G2: Every two distinct points, A and B, lie on a unique line, AB. G3: If lines AB and CD intersect, then so do lines AC and BD (where it is assumed that A and D are distinct from B and C).
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 ...