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.
Famous quotes containing the words concurrent and/or lines:
“I have been too long acquainted with human nature to have great regard for human testimony; and a very great degree of probability, supported by various concurrent circumstances, conspiring in one point, will have much greater weight with me, than human testimony upon oath, or even upon honour; both of which I have frequently seen considerably warped by private views.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)
“Indeed, I believe that in the future, when we shall have seized again, as we will seize if we are true to ourselves, our own fair part of commerce upon the sea, and when we shall have again our appropriate share of South American trade, that these railroads from St. Louis, touching deep harbors on the gulf, and communicating there with lines of steamships, shall touch the ports of South America and bring their tribute to you.”
—Benjamin Harrison (1833–1901)