Search results
Results from the WOW.Com Content Network
Concurrent lines arise in the dual of Pappus's hexagon theorem. For each side of a cyclic hexagon, extend the adjacent sides to their intersection, forming a triangle exterior to the given side. Then the segments connecting the circumcenters of opposite triangles are concurrent.
Given a set of collinear points, by plane duality we obtain a set of lines all of which meet at a common point. The property that this set of lines has (meeting at a common point) is called concurrency, and the lines are said to be concurrent lines. Thus, concurrency is the plane dual notion to collinearity.
Line a is a great circle, the equivalent of a straight line in spherical geometry. Line c is equidistant to line a but is not a great circle. It is a parallel of latitude. Line b is another geodesic which intersects a in two antipodal points. They share two common perpendiculars (one shown in blue).
Concurrent lines, in geometry, multiple lines or curves intersecting at a single point; Concurrency (road), an instance of one physical road bearing two or more different route numbers; Concurrent (Easter), the weekday of 24 March Julian used to calculate Julian Easter
This occurs if the lines are parallel, or if they intersect each other. Two lines that are not coplanar are called skew lines . Distance geometry provides a solution technique for the problem of determining whether a set of points is coplanar, knowing only the distances between them.
If lines Aa, Bb and Cc are concurrent (meet at a point), then the points AB ∩ ab, AC ∩ ac and BC ∩ bc are collinear. The points A, B, a and b are coplanar (lie in the same plane) because of the assumed concurrency of Aa and Bb. Therefore, the lines AB and ab belong to the same plane and must intersect.
Variants of this concept exist for a quadruple of concurrent lines on the projective plane and a quadruple of points on the Riemann sphere. In the Cayley–Klein model of hyperbolic geometry, the distance between points is expressed in terms of a certain cross-ratio.
Then the three lines A ′ A ′ ′, B ′ B ′ ′, C ′ C ′ ′, are concurrent. [2] The point at which they concur is the orthopole. Due to their many properties, [3] orthopoles have been the subject of a large literature. [4] Some key topics are determination of the lines having a given orthopole [5] and orthopolar circles. [6]