In geometry, a cevian is a line segment which joins a vertex of a triangle to a point on the opposite side of the triangle.^[1][2] 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.^[3]

There are various properties of the ratios of lengths formed by three cevians all passing through the same arbitrary interior point:^{[6]: 177–188} Referring to the diagram at right,

${\displaystyle {\begin{aligned}&{\frac {\overline {AF}}{\overline {FB}}}\cdot {\frac {\overline {BD}}{\overline {DC}}}\cdot {\frac {\overline {CE}}{\overline {EA}}}=1\\&\\&{\frac {\overline {AO}}{\overline {OD}}}={\frac {\overline {AE}}{\overline {EC}}}+{\frac {\overline {AF}}{\overline {FB}}};\\&\\&{\frac {\overline {OD}}{\overline {AD}}}+{\frac {\overline {OE}}{\overline {BE}}}+{\frac {\overline {OF}}{\overline {CF}}}=1;\\&\\&{\frac {\overline {AO}}{\overline {AD}}}+{\frac {\overline {BO}}{\overline {BE}}}+{\frac {\overline {CO}}{\overline {CF}}}=2.\end{aligned}}}$

The first property is known as Ceva's theorem. The last two properties are equivalent because summing the two equations gives the identity 1 + 1 + 1 = 3.

A splitter of a triangle is a cevian that bisects the perimeter. The three splitters concur at the Nagel point of the triangle.

Three of the area bisectors of a triangle are its medians, which connect the vertices to the opposite side midpoints. Thus a uniform-density triangle would in principle balance on a razor supporting any of the medians.

If from each vertex of a triangle two cevians are drawn so as to trisect the angle (divide it into three equal angles), then the six cevians intersect in pairs to form an equilateral triangle, called the Morley triangle.

Routh's theorem determines the ratio of the area of a given triangle to that of a triangle formed by the pairwise intersections of three cevians, one from each vertex.

• Mass point geometry
• Menelaus' theorem