Concurrent lines

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


angle's altitudes run from each vertex and meet the opposite side at a right angle. The point where the three altitudes meet is the orthocenter.

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



Regular polygons






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.


  1. Dunn, J. A., and Pretty, J. E., "Halving a triangle," Mathematical Gazette 56, May 1972, 105-108.
  2. Kodokostas, Dimitrios, "Triangle Equalizers," Mathematics Magazine 83, April 2010, pp. 141-146.
  3. 1 2 Altshiller-Court, Nathan (2007) [1952], College Geometry: An Introduction to the Modern Geometry of the Triangle and the Circle (2nd ed.), Courier Dover, pp. 131, 137–8, ISBN 978-0-486-45805-2, OCLC 78063045
  4. Andreescu, Titu and Enescu, Bogdan, Mathematical Olympiad Treasures, Birkhäuser, 2006, pp. 64–68.
  5. Honsberger, Ross (1995), "4.2 Cyclic quadrilaterals", Episodes in Nineteenth and Twentieth Century Euclidean Geometry, New Mathematical Library, 37, Cambridge University Press, pp. 35–39, ISBN 978-0-88385-639-0
  6. Weisstein, Eric W. "Maltitude". MathWorld.
  7. Cartensen, Jens, "About hexagons", Mathematical Spectrum 33(2) (2000-2001), 37-40.
  8. Nikolaos Dergiades, "Dao's theorem on six circumcenters associated with a cyclic hexagon", Forum Geometricorum 14, 2014, 243--246.
  9. Leung, Kam-tim; and Suen, Suk-nam; "Vectors, matrices and geometry", Hong Kong University Press, 1994, pp. 53-54

External links

This article is issued from Wikipedia - version of the 11/3/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.