Cevian

In geometry, a cevian is any line segment in a triangle with one endpoint on a vertex of the triangle and the other endpoint on the opposite side.Medians, altitudes, and angle bisectors are special cases of cevians. The name cevian comes from the Italian engineer Giovanni Ceva, who proved a well-known theorem about cevians which also bears his name.

The length of a cevian can be determined by Stewart's theorem: in the diagram, the cevian length d is given by the formula

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

or

since

Hence in this case

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

and

and

where the semiperimeter s = (a+b+c)/2.

The side of length a is divided in the proportion b:c.

If the cevian happens to be an altitude and thus perpendicular to a side, its length obeys the formulas

and

where the semiperimeter s = (a+b+c) / 2.

There are various properties of the ratios of lengths formed by three cevians all passing through the same arbitrary interior point: Referring to the diagram at right,

These 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.

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