In geometry

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 ....

, a cevian is any line segment

Line segment

In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. Examples of line segments include the sides of a triangle or square. More generally, when the end points are both vertices of a polygon, the line segment...

 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 ....

 with one endpoint on a vertex

Vertex (geometry)

In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...

 of the triangle and the other endpoint on the opposite side. 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...

, 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...

, and angle bisectors are special cases of cevians. The name cevian comes from the Italian engineer Giovanni Ceva

Giovanni Ceva

Giovanni Ceva was an Italian mathematician widely known for proving Ceva's theorem in elementary geometry. His brother, Tommaso Ceva was also a well known poet and mathematician....

, who proved a well known theorem about cevians which also bears his name.

Length

The length of a cevian can be determined by Stewart's Theorem

Stewart's theorem

In geometry, Stewart's theorem yields a relation between a lengths of the sides of the triangle and the length of a cevian of the triangle. Its name is in honor of the Scottish mathematician Matthew Stewart who published the theorem in 1746.- Theorem :...

. In the diagram, the length can be determined from the formula


If the cevian happens to be a median, its length can be determined from the formula


If the cevian happens to be an angle bisector, its length can be determined from the formula


If the cevian happens to be an altitude, its length can be determined from the formula