In
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...
, Ceva's theorem is a theorem about
triangle
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...
s. Given a triangle , let the
lines be drawn from the
vertices to a common point (not on one of the sides of ), to meet opposite sides at respectively. (The segments are known as
cevians.) Then, using
signed lengths of segments,
:
In other words, the length is taken to be positive or negative according to whether is to the left or right of in some fixed orientation of the line. For example, is defined as having positive value when is between and and negative otherwise.
Ceva's theorem is a theorem of
affine geometry
In mathematics, affine geometry is what remains of Euclidean geometry when ignoring (mathematicians often say "forgetting") the metric notions of distance and angle.
As the notion of '' parallel lines'' is one of the main properties that is i ...
, in the sense that it may be stated and proved without using the concepts of angles, areas, and lengths (except for the ratio of the lengths of two
line segment
In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct endpoints (its extreme points), and contains every Point (geometry), point on the line that is between its endpoints. It is a special c ...
s that are
collinear
In geometry, collinearity of a set of Point (geometry), points is the property of their lying on a single Line (geometry), line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, t ...
). It is therefore true for triangles in any
affine plane
In geometry, an affine plane is a two-dimensional affine space.
Definitions
There are two ways to formally define affine planes, which are equivalent for affine planes over a field.
The first way consists in defining an affine plane as a set on ...
over any
field.
A slightly adapted
converse is also true: If points are chosen on respectively so that
:
then are
concurrent, or all three
parallel. The converse is often included as part of the theorem.
The theorem is often attributed to
Giovanni Ceva
Giovanni Ceva (September 1, 1647 – May 13, 1734) was an Italian mathematician widely known for proving Ceva's theorem in elementary geometry. His brother, Tommaso Ceva, was also a well-known poet and mathematician.
Life
Ceva received his edu ...
, who published it in his 1678 work ''De lineis rectis''. But it was proven much earlier by
Yusuf Al-Mu'taman ibn Hűd, an eleventh-century king of
Zaragoza
Zaragoza (), traditionally known in English as Saragossa ( ), is the capital city of the province of Zaragoza and of the autonomous communities of Spain, autonomous community of Aragon, Spain. It lies by the Ebro river and its tributaries, the ...
.
Associated with the figures are several terms derived from Ceva's name:
cevian (the lines are the cevians of ), cevian triangle (the triangle is the cevian triangle of ); cevian nest, anticevian triangle, Ceva conjugate. (''Ceva'' is pronounced Chay'va; ''cevian'' is pronounced chev'ian.)
The theorem is very similar to
Menelaus' theorem
In Euclidean geometry, Menelaus's theorem, named for Menelaus of Alexandria, is a proposition about triangles in plane geometry. Suppose we have a triangle , and a Transversal (geometry), transversal line that crosses at points respectively, wi ...
in that their equations differ only in sign. By re-writing each in terms of
cross-ratios, the two theorems may be seen as
projective duals.
Proofs
Several proofs of the theorem have been created.
Two proofs are given in the following.
The first one is very elementary, using only basic properties of triangle areas.
However, several cases have to be considered, depending on the position of the point .
The second proof uses
barycentric coordinates and
vectors, but is more natural and not case dependent. Moreover, it works in any
affine plane
In geometry, an affine plane is a two-dimensional affine space.
Definitions
There are two ways to formally define affine planes, which are equivalent for affine planes over a field.
The first way consists in defining an affine plane as a set on ...
over any
field.
Using triangle areas
First, the sign of the
left-hand side is positive since either all three of the ratios are positive, the case where is inside the triangle (upper diagram), or one is positive and the other two are negative, the case is outside the triangle (lower diagram shows one case).
To check the magnitude, note that the area of a triangle of a given height is proportional to its base. So
:
Therefore,
:
(Replace the minus with a plus if and are on opposite sides of .)
Similarly,
:
and
:
Multiplying these three equations gives
:
as required.
The theorem can also be proven easily using
Menelaus's theorem. From the transversal of triangle ,
:
and from the transversal of triangle ,
:
The theorem follows by dividing these two equations.
The converse follows as a corollary.
[ Let be given on the lines so that the equation holds. Let meet at and let be the point where crosses . Then by the theorem, the equation also holds for . Comparing the two,
:
But at most one point can cut a segment in a given ratio so .
]
Using barycentric coordinates
Given three points that are not collinear
In geometry, collinearity of a set of Point (geometry), points is the property of their lying on a single Line (geometry), line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, t ...
, and a point , that belongs to the same plane, the barycentric coordinates of with respect of are the unique three numbers such that
:
and
:
for every point (for the definition of this arrow notation and further details, see Affine space
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relat ...
).
For Ceva's theorem, the point is supposed to not belong to any line passing through two vertices of the triangle. This implies that
If one takes for the intersection of the lines and (see figures), the last equation may be rearranged into
:
The left-hand side of this equation is a vector that has the same direction as the line , and the right-hand side has the same direction as the line . These lines have different directions since are not collinear. It follows that the two members of the equation equal the zero vector, and
:
It follows that
:
where the left-hand-side fraction is the signed ratio of the lengths of the collinear line segment
In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct endpoints (its extreme points), and contains every Point (geometry), point on the line that is between its endpoints. It is a special c ...
s and .
The same reasoning shows
:
Ceva's theorem results immediately by taking the product of the three last equations.
Generalizations
The theorem can be generalized to higher-dimensional simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
es using barycentric coordinates. Define a cevian of an -simplex as a ray from each vertex to a point on the opposite ()-face (facet
Facets () are flat faces on geometric shapes. The organization of naturally occurring facets was key to early developments in crystallography, since they reflect the underlying symmetry of the crystal structure. Gemstones commonly have facets cu ...
). Then the cevians are concurrent if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
a mass distribution
In physics and mechanics, mass distribution is the spatial distribution of mass within a solid body. In principle, it is relevant also for gases or liquids, but on Earth their mass distribution is almost homogeneous.
Astronomy
In astronomy mass ...
can be assigned to the vertices such that each cevian intersects the opposite facet at its center of mass
In physics, the center of mass of a distribution of mass in space (sometimes referred to as the barycenter or balance point) is the unique point at any given time where the weight function, weighted relative position (vector), position of the d ...
. Moreover, the intersection point of the cevians is the center of mass of the simplex.
Another generalization to higher-dimensional simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
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 -face. This point is the foot of a cevian that goes from the vertex opposite the -face, in a ()-face that contains it, through the point already defined on this ()-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.
Routh's theorem
In geometry, Routh's theorem determines the ratio of areas between a given triangle and a triangle formed by the pairwise intersections of three cevians. The theorem states that if in triangle ABC points D, E, and F lie on segments BC, CA, and ...
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
In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain.
The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...
s in the plane has been known since the early nineteenth century.
The theorem has also been generalized to triangles on other surfaces of constant curvature
In mathematics, constant curvature is a concept from differential geometry. Here, curvature refers to the sectional curvature of a space (more precisely a manifold) and is a single number determining its local geometry. The sectional curvature is ...
.
The theorem also has a well-known generalization to spherical and hyperbolic geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For a ...
, replacing the lengths in the ratios with their sines and hyperbolic sines, respectively.
See also
*Projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting (''p ...
*Median (geometry)
In geometry, a median of a triangle is a line segment joining a vertex to the midpoint of the opposite side, thus bisecting that side. Every triangle has exactly three medians, one from each vertex, and they all intersect at the triangle's cent ...
– an application
* Circumcevian triangle
* Menelaus's theorem
*Triangle
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...
*Stewart's theorem
In geometry, Stewart's theorem yields a relation between the lengths of the sides and the length of a cevian in a triangle. Its name is in honour of the Scottish mathematician Matthew Stewart, who published the theorem in 1746.
Statement
Let ...
* Cevian
References
Further reading
*
External links
Menelaus and Ceva
at MathPages
Derivations and applications of Ceva's Theorem
at cut-the-knot
Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet Union, Soviet-born Israeli Americans, Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow ...
Trigonometric Form of Ceva's Theorem
at cut-the-knot
Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet Union, Soviet-born Israeli Americans, Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow ...
Glossary of Encyclopedia of Triangle Centers
includes definitions of cevian triangle, cevian nest, anticevian triangle, Ceva conjugate, and cevapoint
Conics Associated with a Cevian Nest, by Clark Kimberling
*'
Ceva's Theorem
' by Jay Warendorff, Wolfram Demonstrations Project
The Wolfram Demonstrations Project is an Open source, open-source collection of Interactive computing, interactive programmes called Demonstrations. It is hosted by Wolfram Research. At its launch, it contained 1300 demonstrations but has grown t ...
.
*
Experimentally finding the centroid of a triangle with different weights at the vertices: a practical application of Ceva's theorem
a
an interactive dynamic geometry sketch using the gravity simulator of Cinderella.
*
{{DEFAULTSORT:Ceva's Theorem
Affine geometry
Theorems about triangles
Articles containing proofs
Euclidean plane geometry