Concurrent Lines

In geometry, three or more lines in a plane 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.

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