Line segment joining the vertex of a triangle to its opposite side.

### Properties

In triangles, altitudes, medians and angle bisectors are special types of cevians.

- The length of a cevian can be determined by using the formula:

\(b^{2}m+c^{2}n = a(d^{2}+mn)\)

- If the cevian is an
*altitude*, its length is given by the formula:

\(d^{2}=b^{2}-n^{2}=c^{2}-m^{2}\). - If the cevian is a
*median*, its length is given by the formula:

\(m(b^{2}+c^{2})=a(d^{2}+m^{2})\). - If the cevian is a
*bisector*, its length is given by the formula:

\({(b+c)}^{2}=a^{2}(\frac{d^{2}}{mn}+1)\).

### Historical Note

The term *cervian* is named for Italian mathematician Giovanni Ceva, who proved a theorem about cevians that also bears his name.