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 ...
, 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
A triangle is a polygon with three edges and three 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, any three points, when non- colline ...
points
,
, and
lie on segments
,
, and
, then writing
,
, and
, the signed
area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an op ...
of the triangle formed by the cevians
,
, and
is
:
where
is the area of the triangle
.
This theorem was given by
Edward John Routh
Edward John Routh (; 20 January 18317 June 1907), was an English mathematician, noted as the outstanding coach of students preparing for the Mathematical Tripos examination of the University of Cambridge in its heyday in the middle of the ninet ...
on page 82 of his ''Treatise on Analytical Statics with Numerous Examples'' in 1896. The particular case
has become popularized as the
one-seventh area triangle. The
case implies that the three
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 f ...
s are concurrent (through the
centroid
In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ...
).
Proof
Suppose that the area of triangle
is 1. For triangle
and line
using
Menelaus's 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'' respe ...
, We could obtain:
:
Then
So the area of triangle
is:
:
Similarly, we could know:
and
Thus the area of triangle
is:
:
Citations
The citation commonly given for Routh's theorem is Routh's ''Treatise on Analytical Statics with Numerous Examples'', Volume 1, Chap. IV, in th
second editionof 189
p. 82 possibly because that edition has been easier to hand. However, Routh gave the theorem already in th
first editionof 1891, Volume 1, Chap. IV
p. 89 Although there is a change in pagination between the editions, the wording of the relevant footnote remained the same.
Routh concludes his extended footnote with a ''caveat'':
"The author has not met with these expressions for the areas of two triangles that often occur. He has therefore placed them here in order that the argument in the text may be more easily understood."
Presumably, Routh felt those circumstances had not changed in the five years between editions. On the other hand, the title of Routh's book had been used earlier by
Isaac Todhunter
Isaac Todhunter FRS (23 November 1820 – 1 March 1884), was an English mathematician who is best known today for the books he wrote on mathematics and its history.
Life and work
The son of George Todhunter, a Nonconformist minister, a ...
; both had been coached by
William Hopkins
William Hopkins FRS (2 February 179313 October 1866) was an English mathematician and geologist. He is famous as a private tutor of aspiring undergraduate Cambridge mathematicians, earning him the ''sobriquet'' the " senior-wrangler maker."
...
.
Although Routh published the theorem in his book, that is not the first published statement. It is stated and proved as rider (vii) on page 33 of Solutions of the Cambridge Senate-house Problems and Riders for the Year 1878, i.e., the mathematical tripos of that year, and the link is https://archive.org/details/solutionscambri00glaigoog. It is stated that the author of the problems with roman numerals is
Glaisher.
Routh was a famous
Mathematical Tripos
The Mathematical Tripos is the mathematics course that is taught in the Faculty of Mathematics at the University of Cambridge. It is the oldest Tripos examined at the University.
Origin
In its classical nineteenth-century form, the tripos was ...
coach when his book came out and was surely familiar with the content of the 1878 tripos examination. Thus, his statement ''The author has not met with these expressions for the areas of two triangles that often occur. '' is puzzling.
Problems in this spirit have a long history in
recreational mathematics
Recreational mathematics is mathematics carried out for recreation (entertainment) rather than as a strictly research and application-based professional activity or as a part of a student's formal education. Although it is not necessarily limited ...
and mathematical
paedagogy
Pedagogy (), most commonly understood as the approach to teaching, is the theory and practice of learning, and how this process influences, and is influenced by, the social, political and psychological development of learners. Pedagogy, taken a ...
, perhaps one of the oldest instances of being the determination of the proportions of the fourteen regions of the
Stomachion board. With Routh's
Cambridge
Cambridge ( ) is a university city and the county town in Cambridgeshire, England. It is located on the River Cam approximately north of London. As of the 2021 United Kingdom census, the population of Cambridge was 145,700. Cambridge bec ...
in mind, the ''
one-seventh-area triangle'', associated in some accounts with
Richard Feynman
Richard Phillips Feynman (; May 11, 1918 – February 15, 1988) was an American theoretical physicist, known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of the superfl ...
, shows up, for example, as Question 100
p. 80 in ''Euclid's Elements of Geometry
Fifth School Edition'', by
Robert Potts (1805--1885,) of Trinity College, published in 1859; compare also his Questions 98, 99, on the same page. Potts stood twenty-sixth
Wrangler in 1832 and then, like Hopkins and Routh, coached at Cambridge. Pott's expository writings in geometry were recognized by
medalat the International Exhibition of 1862, as well as by an Hon. LL.D. from the
College of William and Mary
The College of William & Mary (officially The College of William and Mary in Virginia, abbreviated as William & Mary, W&M) is a public research university in Williamsburg, Virginia. Founded in 1693 by letters patent issued by King William ...
,
Williamsburg,
Virginia
Virginia, officially the Commonwealth of Virginia, is a state in the Mid-Atlantic and Southeastern regions of the United States, between the Atlantic Coast and the Appalachian Mountains. The geography and climate of the Commonwealth are ...
.
References
*
Murray S. Klamkin and A. Liu (1981) "Three more proofs of Routh's theorem", ''
Crux Mathematicorum
''Crux Mathematicorum'' is a scientific journal of mathematics published by the Canadian Mathematical Society. It contains mathematical problems for secondary school and undergraduate students. , its editor-in-chief is Kseniya Garaschuk.
The journ ...
'' 7:199–203.
*
H. S. M. Coxeter
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington t ...
(1969) ''Introduction to Geometry'', statement p. 211, proof pp. 219–20, 2nd edition, Wiley, New York.
* J. S. Kline and D. Velleman (1995) "Yet another proof of Routh's theorem" (1995) ''
Crux Mathematicorum
''Crux Mathematicorum'' is a scientific journal of mathematics published by the Canadian Mathematical Society. It contains mathematical problems for secondary school and undergraduate students. , its editor-in-chief is Kseniya Garaschuk.
The journ ...
'' 21:37–40
*
Ivan Niven (1976) "A New Proof of Routh's Theorem",
Mathematics Magazine
''Mathematics Magazine'' is a refereed bimonthly publication of the Mathematical Association of America. Its intended audience is teachers of collegiate mathematics, especially at the junior/senior level, and their students. It is explicitly a j ...
49(1): 25–7,
* Jay Warendorff
Routh's Theorem The Wolfram Demonstrations Project.
* {{MathWorld , title=Routh's Theorem , urlname=RouthsTheorem
Routh's Theorem by Cross Productsat MathPages
* Ayoub, Ayoub B. (2011/2012) "Routh's theorem revisited", ''Mathematical Spectrum'' 44 (1): 24-27.
Theorems about triangles
Area
Affine geometry