Stewart's Theorem
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In
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, 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 , , be the lengths of the sides of a triangle. Let be the length of a cevian 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 b^2m + c^2n = a(d^2 + mn). A common
mnemonic A mnemonic device ( ), memory trick or memory device is any learning technique that aids information retention or retrieval in the human memory, often by associating the information with something that is easier to remember. It makes use of e ...
used by students to memorize this equation (after rearranging the terms) is: \underset = \!\!\!\!\!\! \underset 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 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 ...
points, and is any point, then :\left(\overline^2\cdot \overline\right) + \left(\overline^2\cdot \overline\right) + \left(\overline^2\cdot \overline\right) + \left(\overline\cdot \overline\cdot \overline\right) =0. In the special case where the cevian is a
median The median of a set of numbers is the value separating the higher half from the lower half of a Sample (statistics), data sample, a statistical population, population, or a probability distribution. For a data set, it may be thought of as the “ ...
(meaning 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, toge ...
.


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 or cosine rule) relates the lengths of the sides of a triangle to the cosine of one of its angles. For a triangle with sides , , and , opposite respective angles , , and (see ...
. Let be the angle between and and the
angle In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...
between and . Then is the supplement of , and so . Applying the law of cosines in the two small triangles using angles and produces \begin c^2 &= m^2 + d^2 - 2dm\cos\theta, \\ b^2 &= n^2 + d^2 - 2dn\cos\theta' \\ &= n^2 + d^2 + 2dn\cos\theta. \end Multiplying the first equation by and the third equation by and adding them eliminates . One obtains \begin b^2m + c^2n &= nm^2 + n^2m + (m+n)d^2 \\ &= (m+n)(mn + d^2) \\ &= a(mn + d^2), \\ \end 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 In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
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 (; ; 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 being the youngest professor. ...
as Professor of Mathematics at the University of Edinburgh. state that the result was probably known to
Archimedes Archimedes of Syracuse ( ; ) was an Ancient Greece, Ancient Greek Greek mathematics, mathematician, physicist, engineer, astronomer, and Invention, inventor from the ancient city of Syracuse, Sicily, Syracuse in History of Greek and Hellenis ...
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 Lazare Nicolas Marguerite, Comte Carnot (; 13 May 1753 – 2 August 1823) was a French mathematician, physicist, military officer, politician and a leading member of the Committee of Public Safety during the French Revolution. His military refor ...
in 1803.


See also

* Mass point geometry


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