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The lines connecting the feet of the altitudes intersect the opposite sides at collinear points. [3]: p.199 A triangle's incenter, the midpoint of an altitude, and the point of contact of the corresponding side with the excircle relative to that side are collinear. [4]: p.120, #78
Tangential – intersecting a curve at a point and parallel to the curve at that point. Collinear – in the same line; Parallel – in the same direction. Transverse – intersecting at any angle, i.e. not parallel. Orthogonal (or perpendicular) – at a right angle (at the point of intersection).
In the case of a line arrangement, each coordinate of the labeling assigns 0 to nodes on one side of one of the lines and 1 to nodes on the other side. [26] Dual graphs of simplicial arrangements have been used to construct infinite families of 3-regular partial cubes, isomorphic to the graphs of simple zonohedra. [27]
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 the circumcircle of the
If the conic is a circle, then another degenerate case says that for a triangle, the three points that appear as the intersection of a side line with the corresponding side line of the Gergonne triangle, are collinear. Six is the minimum number of points on a conic about which special statements can be made, as five points determine a conic.
The line AB is the interval AB and the two rays A/B and B/A. Points on the line AB are said to be collinear. An angle consists of a point O (the vertex) and two non-collinear rays out from O (the sides). A triangle is given by three non-collinear points (called vertices) and their three segments AB, BC, and CA.
In Euclidean and projective geometry, five points determine a conic (a degree-2 plane curve), just as two (distinct) points determine a line (a degree-1 plane curve).There are additional subtleties for conics that do not exist for lines, and thus the statement and its proof for conics are both more technical than for lines.
One also says "two generic lines intersect in a point", which is formalized by the notion of a generic point. 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 .