geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, a cevian is a

line Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Art ...

that intersects both a

triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colli ...

's vertex, and also the side that is opposite to that vertex. Medians and angle bisectors are special cases of cevians. The name "cevian" comes from the Italian mathematician Giovanni Ceva, who proved a well-known theorem about cevians which also bears his name.

Length

Stewart's theorem

The length of a cevian can be determined by

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

: in the diagram, the cevian length is given by the formula :

\,b^2m + c^2n = a(d^2 + mn).

Less commonly, this is also represented (with some rearrangement) by the following

mnemonic A mnemonic ( ) device, or memory device, is any learning technique that aids information retention or retrieval (remembering) in the human memory for better understanding. Mnemonics make use of elaborative encoding, retrieval cues, and image ...

: :

\underset = \!\!\!\!\!\! \underset

Median

If the cevian happens to be a median (thus bisecting a side), its length can be determined from the formula :

\,m(b^2 + c^2) = a(d^2 + m^2)

or :

\,2(b^2 + c^2) = 4d^2 + a^2

since :

\,a = 2m.

Hence in this case :

d= \frac\sqrt2 .

Angle bisector

If the cevian happens to be an angle bisector, its length obeys the formulas