Ceva theorem

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A theorem on the relation between the lengths of certain lines intersecting a triangle. Let $A_1,B_1,C_1$ be three points lying, respectively, on the sides $BC$, $CA$ and $AB$ of a triangle $ABC$. For the lines $AA_1$, $BB_1$ and $CC_1$ to intersect in a single point or to be all parallel it is necessary and sufficient that


Lines $AA_1$, $BB_1$ and $CC_1$ 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 $0$ be given in a planar polygon with an odd number of vertices $A_1\dots A_{2n-1}$, and suppose that the lines $0A_1,\dots,0A_{2n-1}$ intersect the sides of the polygon opposite to $A_1,\dots,A_{2n-1}$ respectively in points $a_n,\dots,a_{2n-1}$, $a_1,\dots,a_{n-1}$. In this case



[1] G. Ceva, "De lineis rectis se invicem secantibus statica constructio" , Milano (1678)
[a1] M. Berger, "Geometry" , I , Springer (1987)
How to Cite This Entry:
Ceva theorem. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by P.S. Modenov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article