# Cevian

## Cevian

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.