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# 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.

Undergraduate Texts in Mathematics

Graduate Studies in Mathematics

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