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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.

Length

A triangle with a cevian

The length of a cevian can be determined by Stewart's Theorem. In the diagram, the length d can be determined from the formula

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

If the cevian happens to be a median, its length can be determined from the formula

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

If the cevian happens to be an angle bisector, its length can be determined from the formula

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

If the cevian happens to be an altitude, its length can be determined from the formula

\( \,d^2 = b^2 - n^2 = c^2 - m^2. \)

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Cevian