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: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...
, Ceva's theorem is a theorem about
triangle
A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, an ...
s. Given a triangle , let the
lines
Line most often refers to:
* Line (geometry), object with zero thickness and curvature that stretches to infinity
* Telephone line, a single-user circuit on a telephone communication system
Line, lines, The Line, or LINE may also refer to:
Arts ...
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
cevian
In geometry, a cevian is a line that intersects both a triangle's vertex, and also the side that is opposite to that vertex. Medians and angle bisectors are special cases of cevians. The name "cevian" comes from the Italian mathematician Giovann ...
s.) 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 inde ...
, 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 straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...
s that are
collinear
In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned ...
). It is therefore true for triangles in any
affine plane
In geometry, an affine plane is a two-dimensional affine space.
Examples
Typical examples of affine planes are
* Euclidean planes, which are affine planes over the reals equipped with a metric, the Euclidean distance. In other words, an affine pl ...
over any
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
.
A slightly adapted
converse
Converse may refer to:
Mathematics and logic
* Converse (logic), the result of reversing the two parts of a definite or implicational statement
** Converse implication, the converse of a material implication
** Converse nonimplication, a logical c ...
is also true: If points are chosen on respectively so that
:
then are
concurrent
Concurrent means happening at the same time. Concurrency, concurrent, or concurrence may refer to:
Law
* Concurrence, in jurisprudence, the need to prove both ''actus reus'' and ''mens rea''
* Concurring opinion (also called a "concurrence"), a ...
, or all three
parallel
Parallel is a geometric term of location which may refer to:
Computing
* Parallel algorithm
* Parallel computing
* Parallel metaheuristic
* Parallel (software), a UNIX utility for running programs in parallel
* Parallel Sysplex, a cluster of ...
. 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 educa ...
, 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, also known in English as Saragossa,''Encyclopædia Britannica'"Zaragoza (conventional Saragossa)" is the capital city of the Zaragoza Province and of the autonomous community of Aragon, Spain. It lies by the Ebro river and its tributari ...
.
Associated with the figures are several terms derived from Ceva's name:
cevian
In geometry, a cevian is a line that intersects both a triangle's vertex, and also the side that is opposite to that vertex. Medians and angle bisectors are special cases of cevians. The name "cevian" comes from the Italian mathematician Giovann ...
(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
Menelaus's theorem, named for Menelaus of Alexandria, is a proposition about triangles in plane geometry. Suppose we have a triangle ''ABC'', and a transversal line that crosses ''BC'', ''AC'', and ''AB'' at points ''D'', ''E'', and ''F'' respec ...
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 given.
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
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 related ...
and
vectors, but is somehow more natural and not case dependent. Moreover, it works in any
affine plane
In geometry, an affine plane is a two-dimensional affine space.
Examples
Typical examples of affine planes are
* Euclidean planes, which are affine planes over the reals equipped with a metric, the Euclidean distance. In other words, an affine pl ...
over any
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
.
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' 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 points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned ...
, and a point , that belongs to the same plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...
, the barycentric coordinates
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 related ...
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 relate ...
).
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 straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...
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
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 related ...
. 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 cut ...
). 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
In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
. 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 that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...
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 i ...
.
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.
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, pro ...
*Median (geometry)
In geometry, a median of a triangle is a line segment joining a vertex (geometry), 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 each ...
– an application
*Circumcevian triangle In triangle geometry, a circumcevian triangle is a special triangle associated with the reference triangle and a point in the plane of the triangle. It is also associated with the circumcircle of the reference triangle. Definition
Let P be a point ...
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-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...
Trigonometric Form of Ceva's Theorem
at cut-the-knot
Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...
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 organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hos ...
.
*
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