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 (extended) opposite side. Medians, altitudes, 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.
The length of a cevian can be determined by Stewart's theorem: in the diagram, the cevian length d is given by the formula
or, less commonly,
Hence in this case
If the cevian happens to be an angle bisector, its length obeys the formulas
where the semiperimeter s = (a+b+c)/2.
The side of length a is divided in the proportion b:c.
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::177-188 Referring to the diagram at right,
These last two properties are equivalent because summing the two equations gives the identity 1 + 1 + 1 = 3.
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.
Area of inner triangle formed by cevians
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.
- Coxeter, H. S. M.; Greitzer, S. L. (1967). Geometry Revisited. Washington, DC: Mathematical Association of America. p. 4. ISBN 0-883-85619-0.
- Lightner, James E. (1975). "A new look at the 'centers' of a triangle". The Mathematics Teacher. 68 (7): 612–615. JSTOR 27960289.
- "Art of Problem Solving". artofproblemsolving.com. Retrieved 2018-10-22.
- Johnson, Roger A., Advanced Euclidean Geometry, Dover Publ., 2007 (orig. 1929), p. 70.
- Alfred S. Posamentier and Charles T. Salkind, Challenging Problems in Geometry, Dover Publishing Co., second revised edition, 1996.
- Ross Honsberger (1995). Episodes in Nineteenth and Twentieth Century Euclidean Geometry, pages 13 and 137. Mathematical Association of America.
- Vladimir Karapetoff (1929). "Some properties of correlative vertex lines in a plane triangle." American Mathematical Monthly 36: 476–479.
- Indika Shameera Amarasinghe (2011). “A New Theorem on any Right-angled Cevian Triangle.” Journal of the World Federation of National Mathematics Competitions, Vol 24 (02), pp. 29–37.