Ceva theorem
A theorem on the relation between the lengths of certain lines intersecting a triangle. Let be three points lying, respectively, on the sides
,
and
of a triangle
. For the lines
,
and
to intersect in a single point or to be all parallel it is necessary and sufficient that
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Lines ,
and
that meet in a single point and pass through the vertices of a triangle are called Ceva, or Cevian, lines. Ceva's theorem is metrically dual to the Menelaus theorem. It is named after G. Ceva [1].
Ceva's theorem can be generalized to the case of a polygon. Let a point be given in a planar polygon with an odd number of vertices
, and suppose that the lines
intersect the sides of the polygon opposite to
respectively in points
,
. In this case
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References
[1] | G. Ceva, "De lineis rectis se invicem secantibus statica constructio" , Milano (1678) |
Comments
References
[a1] | M. Berger, "Geometry" , I , Springer (1987) |
Ceva theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ceva_theorem&oldid=17362