In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a cevian is a
line that intersects both a
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
vertex
Vertex, vertices or vertexes may refer to:
Science and technology Mathematics and computer science
*Vertex (geometry), a point where two or more curves, lines, or edges meet
* Vertex (computer graphics), a data structure that describes the positio ...
, and also the side that is opposite to that vertex.
Medians
The Medes ( Old Persian: ; Akkadian: , ; Ancient Greek: ; Latin: ) were an ancient Iranian people who spoke the Median language and who inhabited an area known as Media between western and northern Iran. Around the 11th century BC, th ...
and
angle bisectors
In geometry, bisection is the division of something into two equal or congruent parts, usually by a line, which is then called a ''bisector''. The most often considered types of bisectors are the ''segment bisector'' (a line that passes throug ...
are special cases of cevians. The name "cevian" comes from the Italian mathematician
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 proved a
well-known theorem about cevians which also bears his name.
Length
Stewart's theorem
The length of a cevian can be determined by
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 ...
: in the diagram, the cevian length is given by the formula
:
Less commonly, this is also represented (with some rearrangement) by the following
mnemonic
A mnemonic ( ) device, or memory device, is any learning technique that aids information retention or retrieval (remembering) in the human memory for better understanding.
Mnemonics make use of elaborative encoding, retrieval cues, and imag ...
:
:
Median
If the cevian happens to be a
median
In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic fe ...
(thus
bisecting a side), its length can be determined from the formula
:
or
:
since
:
Hence in this case
:
Angle bisector
If the cevian happens to be an
angle bisector
In geometry, bisection is the division of something into two equal or congruent parts, usually by a line, which is then called a ''bisector''. The most often considered types of bisectors are the ''segment bisector'' (a line that passes through ...
, its length obeys the formulas
:
and
[Johnson, Roger A., ''Advanced Euclidean Geometry'', Dover Publ., 2007 (orig. 1929), p. 70.]
:
and
:
where the
semiperimeter
In geometry, the semiperimeter of a polygon is half its perimeter. Although it has such a simple derivation from the perimeter, the semiperimeter appears frequently enough in formulas for triangles and other figures that it is given a separate name ...
The side of length is divided in the proportion .
Altitude
If the cevian happens to be an
altitude
Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...
and thus
perpendicular
In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...
to a side, its length obeys the formulas
:
and
:
where the semiperimeter
Ratio properties
![Ceva's theorem 1](https://upload.wikimedia.org/wikipedia/commons/c/c7/Ceva%27s_theorem_1.svg)
There are various properties of the ratios of lengths formed by three cevians all passing through the same arbitrary interior point:
[ Alfred S. Posamentier and Charles T. Salkind, ''Challenging Problems in Geometry'', Dover Publishing Co., second revised edition, 1996.] Referring to the diagram at right,
:
The first property is known as
Ceva's theorem
In Euclidean geometry, Ceva's theorem is a theorem about triangles. 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 kn ...
. The last two properties are equivalent because summing the two equations gives the
identity
Identity may refer to:
* Identity document
* Identity (philosophy)
* Identity (social science)
* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film), ...
.
Splitter
A
splitter of a triangle is a cevian that
bisects the
perimeter
A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference.
Calculating the perimeter has several pract ...
. The three splitters
concur
In Western jurisprudence, concurrence (also contemporaneity or simultaneity) is the apparent need to prove the simultaneous occurrence of both ("guilty action") and ("guilty mind"), to constitute a crime; except in crimes of strict liability ...
at the
Nagel point
In geometry, the Nagel point (named for Christian Heinrich von Nagel) is a triangle center, one of the points associated with a given triangle whose definition does not depend on the placement or scale of the triangle. It is the point of concur ...
of the triangle.
Area bisectors
Three of the
area bisectors of a triangle are its medians, which connect the vertices to the opposite side midpoints. Thus a uniform-density triangle would in principle balance on a razor supporting any of the medians.
Angle trisectors
If from each vertex of a triangle two cevians are drawn so as to
trisect the angle (divide it into three equal angles), then the six cevians intersect in pairs to form an
equilateral triangle
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each othe ...
, called the
Morley triangle
In plane geometry, Morley's trisector theorem states that in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle, called the first Morley triangle or simply the Morley triangle. The theorem ...
.
Area of inner triangle formed by cevians
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 ...
determines the ratio of the area of a given triangle to that of a triangle formed by the pairwise intersections of three cevians, one from each vertex.
See also
*
Mass point geometry
Mass point geometry, colloquially known as mass points, is a problem-solving technique in geometry which applies the physical principle of the center of mass to geometry problems involving triangles and intersecting cevians. All problems that ca ...
*
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 ...
Notes
References
* {{citation, first=Howard, last=Eves, title=A Survey of Geometry (Vol. One), publisher=Allyn and Bacon, year=1963
* Ross Honsberger (1995). ''Episodes in Nineteenth and Twentieth Century Euclidean Geometry'', pages 13 and 137. Mathematical Association of America.
*
Vladimir Karapetoff
Vladimir Karapetoff (January 8, 1876 in Saint Petersburg, Russian Empire – January 11, 1948) was a Russian-American electrical engineer, inventor, professor, and author.
Life
He was the son of Nikita Ivanovich Karapetov and Anna Joakimovna Kara ...
(1929). "Some properties of correlative vertex lines in a plane triangle." ''American Mathematical Monthly'' 36: 476–479.
* Indika Shameera Amarasinghe (2011). “A New Theorem on any Right-angled Cevian Triangle.” ''Journal of the World Federation of National Mathematics Competitions'', Vol 24 (02), pp. 29–37.
Straight lines defined for a triangle