Concurrent lines

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

In any affine space (including a Euclidean space) the set of lines parallel to a given line (sharing the same direction) 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: 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.

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.