Editing Ceva's theorem (section)

==Generalizations==
The theorem can be generalized to higher-dimensional [[simplex]]es using [[Barycentric coordinates (mathematics)|barycentric coordinates]]. Define a cevian of an {{mvar|n}}-simplex as a ray from each vertex to a point on the opposite ({{math|''n'' − 1}})-face ([[Facet (mathematics)|facet]]).  Then the cevians are concurrent [[if and only if]] a [[mass distribution]] can be assigned to the vertices such that each cevian intersects the opposite facet at its [[center of mass]]. Moreover, the intersection point of the cevians is the center of mass of the simplex.<ref>{{cite journal | last1 = Landy | first1 = Steven |date=December 1988 | title = A Generalization of Ceva's Theorem to Higher Dimensions |  journal = The American Mathematical Monthly | volume = 95 | issue = 10| pages = 936–939 | doi = 10.2307/2322390 | jstor = 2322390 }}</ref><ref>{{cite journal | last1 = Wernicke | first1 = Paul |date=November 1927 | title = The Theorems of Ceva and Menelaus and Their Extension |  journal = The American Mathematical Monthly | volume = 34 | issue = 9| pages = 468–472 | doi = 10.2307/2300222| jstor = 2300222 }}</ref>

Another generalization to higher-dimensional [[simplex]]es extends the conclusion of Ceva's theorem that the product of certain ratios is 1. Starting from a point in a simplex, a point is defined inductively on each {{mvar|k}}-face. This point is the foot of a cevian that goes from the vertex opposite the {{mvar|k}}-face, in a ({{math|''k'' + 1}})-face that contains it, through the point already defined on this ({{math|''k'' + 1}})-face.  Each of these points divides the face on which it lies into lobes. Given a cycle of pairs of lobes, the product of the ratios of the volumes of the lobes in each pair is 1.<ref>{{cite journal | last1 = Samet | first1 = Dov|date=May 2021 | title = An Extension of Ceva's Theorem to ''n''-Simplices |  journal = The American Mathematical Monthly | volume = 128 | issue = 5| pages = 435–445 | doi = 10.1080/00029890.2021.1896292| s2cid = 233413469}}</ref>   

[[Routh's theorem]] gives the area of the triangle formed by three cevians in the case that they are not concurrent.  Ceva's theorem can be obtained from it by setting the area equal to zero and solving.

The analogue of the theorem for general [[polygon]]s in the plane has been known since the early nineteenth century.<ref>{{Cite journal | doi=10.2307/2690569 | last1=Grünbaum | first1=Branko | last2=Shephard | first2=G. C. | title=Ceva, Menelaus and the Area Principle | year=1995 | journal=Mathematics Magazine | volume=68 | issue=4 | pages=254–268 | jstor=2690569 }}</ref>
The theorem has also been generalized to triangles on other surfaces of [[constant curvature]].<ref>{{cite journal | last1 = Masal'tsev | first1 = L. A. | year = 1994 | title = Incidence theorems in spaces of constant curvature |  journal = Journal of Mathematical Sciences | volume = 72 | issue =4 | pages =3201–3206 |doi= 10.1007/BF01249519 | s2cid = 123870381 }}</ref>

The theorem also has a well-known generalization to spherical and [[hyperbolic geometry]], replacing the lengths in the ratios with their sines and hyperbolic sines, respectively.