- Concurrent lines
-
In geometry, two or more lines are said to be concurrent if they intersect at a single point.
In a triangle, four basic types of sets of concurrent lines are altitudes, angle bisectors, medians, and perpendicular bisectors:
- A triangle'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.
- Angle bisectors are rays running from each vertex of the triangle and bisecting the associated angle. They all meet at the incenter.
- Medians connect each vertex of a triangle to the midpoint of the opposite side. The three medians meet at the centroid.
- Perpendicular bisectors are lines running out of the midpoints of each side of a triangle at 90 degree angles. The three perpendicular bisectors meet at the circumcenter.
Other sets of lines associated with a triangle are concurrent as well. For example, any median (which is necessarily a bisector of the triangle's area) is concurrent with two other area bisectors each of which is parallel to a side.[1]
Compare to collinear. In projective geometry, in two dimensions concurrency is the dual of collinearity; in three dimensions, concurrency is the dual of coplanarity.
References
- ^ Dunn, J. A., and Pretty, J. E., "Halving a triangle," Mathematical Gazette 56, May 1972, 105-108.
External links
- Wolfram MathWorld Concurrent, 2010.
Categories:- Elementary geometry
Wikimedia Foundation. 2010.
