Concurrent lines
Encyclopedia
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:
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
.
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, two or more lines
Line (mathematics)
The notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects...
are said to be concurrent if they intersect
Line-line intersection
In Euclidean geometry, the intersection of a line and a line can be the empty set,a point, ora line. Distinguishing these cases, and finding the intersection point have use, for example, in computer graphics, motion planning, and collision detection....
at a single point
Point (geometry)
In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...
.
In a triangle
Triangle
A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
, four basic types of sets of concurrent lines are altitudes
Altitude (triangle)
In geometry, an altitude of a triangle is a straight line through a vertex and perpendicular to a line containing the base . This line containing the opposite side is called the extended base of the altitude. The intersection between the extended base and the altitude is called the foot of the...
, angle bisectors, medians
Median (geometry)
In geometry, a median of a triangle is a line segment joining a vertex to the midpoint of the opposing side. Every triangle has exactly three medians; one running from each vertex to the opposite side...
, and perpendicular bisectors:
- A triangle's altitudes run from each vertexVertex (geometry)In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...
and meet the opposite side at a right angleRight angleIn geometry and trigonometry, a right angle is an angle that bisects the angle formed by two halves of a straight line. More precisely, if a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles...
. 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 angleAngleIn geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...
. 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 centroidCentroidIn geometry, the centroid, geometric center, or barycenter of a plane figure or two-dimensional shape X is the intersection of all straight lines that divide X into two parts of equal moment about the line. Informally, it is the "average" of all points of X...
.
- 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
Line (geometry)
The notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects...
. In projective geometry
Projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...
, in two dimensions concurrency is the dual
Duality (projective geometry)
A striking feature of projective planes is the "symmetry" of the roles played by points and lines in the definitions and theorems, and duality is the formalization of this metamathematical concept. There are two approaches to the subject of duality, one through language and the other a more...
of collinearity; in three dimensions, concurrency is the dual of coplanarity
Coplanarity
In geometry, a set of points in space is coplanar if all the points lie in the same geometric plane. For example, three distinct points are always coplanar; but a fourth point or more added in space can exist in another plane, incoplanarly....
.
External links
- Wolfram MathWorld Concurrent, 2010.