Lines A, B and C are concurrent in Y.

In geometry, lines in a plane or higher-dimensional space are concurrent if they intersect at a single point.

The set of all lines through a point is called a pencil, and their common intersection is called the vertex of the pencil. In any affine space (including a Euclidean space) the set of lines parallel to a given line (sharing the same orientation) is also called a pencil, and the vertex of each pencil of parallel lines is a distinct point at infinity; including these points results in a projective space in which every pair of lines has an intersection.



In a triangle, four basic types of sets of concurrent lines are altitudes, angle bisectors, medians, and perpendicular bisectors:

Other sets of lines associated with a triangle are concurrent as well. For example:



Regular polygons






See also: Incidence (geometry) § Concurrence

According to the Rouché–Capelli theorem, a system of equations is consistent if and only if the rank of the coefficient matrix is equal to the rank of the augmented matrix (the coefficient matrix augmented with a column of intercept terms), and the system has a unique solution if and only if that common rank equals the number of variables. Thus with two variables the k lines in the plane, associated with a set of k equations, are concurrent if and only if the rank of the k × 2 coefficient matrix and the rank of the k × 3 augmented matrix are both 2. In that case only two of the k equations are independent, and the point of concurrency can be found by solving any two mutually independent equations simultaneously for the two variables.

Projective geometry

In projective geometry, in two dimensions concurrency is the dual of collinearity; in three dimensions, concurrency is the dual of coplanarity.


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