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 ...
, Stewart's theorem yields a relation between the lengths of the sides and the length of a
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 Giovan ...
in a triangle. Its name is in honour of the Scottish mathematician
Matthew Stewart, who published the theorem in 1746.
Statement
![Stewarts theorem](https://upload.wikimedia.org/wikipedia/commons/c/cd/Stewarts_theorem.svg)
Let be the lengths of the sides of a triangle. Let be the length of a
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 Giovan ...
to the side of length . If the cevian divides the side of length into two segments of length and , with adjacent to and adjacent to , then Stewart's theorem states that
:
A common
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 ...
used by students to memorize this equation (after rearranging the terms) is:
:
The theorem may be written more symmetrically using signed lengths of segments. That is, take the length to be positive or negative according to whether is to the left or right of in some fixed orientation of the line. In this formulation, the theorem states that if 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 o ...
points, and is any point, then
:
In the special case that the cevian is the
median (that is, it divides the opposite side into two segments of equal length), the result is known as
Apollonius' theorem
In geometry, Apollonius's theorem is a theorem relating the length of a median of a triangle to the lengths of its sides.
It states that "the sum of the squares of any two sides of any triangle equals twice the square on half the third side, t ...
.
Proof
The theorem can be proved as an application of the
law of cosines
In trigonometry, the law of cosines (also known as the cosine formula, cosine rule, or al-Kashi's theorem) relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states ...
.
Let be the angle between and and the
angle
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles a ...
between and . Then is the
supplement of , and so . Applying the law of cosines in the two small triangles using angles and produces
:
Multiplying the first equation by and the third equation by and adding them eliminates . One obtains
:
which is the required equation.
Alternatively, the theorem can be proved by drawing a perpendicular from the vertex of the triangle to the base and using the
Pythagorean theorem to write the distances in terms of the altitude. The left and right hand sides of the equation then reduce algebraically to the same expression.
History
According to , Stewart published the result in 1746 when he was a candidate to replace
Colin Maclaurin
Colin Maclaurin (; gd, Cailean MacLabhruinn; February 1698 – 14 June 1746) was a Scottish mathematician who made important contributions to geometry and algebra. He is also known for being a child prodigy and holding the record for bei ...
as Professor of Mathematics at the University of Edinburgh. state that the result was probably known to
Archimedes around 300 B.C.E. They go on to say (mistakenly) that the first known proof was provided by R. Simson in 1751. state that the result is used by Simson in 1748 and by Simpson in 1752, and its first appearance in Europe given by
Lazare Carnot in 1803.
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 ...
Notes
References
*
*
*
Further reading
* I.S Amarasinghe, Solutions to the Problem 43.3: Stewart's Theorem (''A New Proof for the Stewart's Theorem using Ptolemy's Theorem''), ''Mathematical Spectrum'', Vol 43(03), pp. 138 – 139, 2011.
*
External links
*
* {{PlanetMath, title=Stewart's Theorem, urlname=StewartsTheorem
Euclidean plane geometry
Theorems about triangles
Articles containing proofs