![Menelaus' theorem 1](https://upload.wikimedia.org/wikipedia/commons/a/a5/Menelaus%27_theorem_1.svg)
Menelaus's theorem, named for
Menelaus of Alexandria
Menelaus of Alexandria (; grc-gre, Μενέλαος ὁ Ἀλεξανδρεύς, ''Menelaos ho Alexandreus''; c. 70 – 140 CE) was a GreekEncyclopædia Britannica "Greek mathematician and astronomer who first conceived and defined a spheric ...
, is a proposition 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 in
plane geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
. Suppose we have a triangle ''ABC'', and a
transversal line that crosses ''BC'', ''AC'', and ''AB'' at points ''D'', ''E'', and ''F'' respectively, with ''D'', ''E'', and ''F'' distinct from ''A'', ''B'', and ''C''. A weak version of the theorem states that
:
where '', AB, '' is taken to be the ordinary length of segment ''AB'': a positive value.
The theorem can be strengthened to a statement about
signed lengths of segments, which provides some additional information about the relative order of collinear points. Here, the length ''AB'' is taken to be positive or negative according to whether ''A'' is to the left or right of ''B'' in some fixed orientation of the line; for example, ''AF''/''FB'' is defined as having positive value when ''F'' is between ''A'' and ''B'' and negative otherwise. The signed version of Menelaus's theorem states
:
Equivalently,
:
Some authors organize the factors differently and obtain the seemingly different relation
:
but as each of these factors is the negative of the corresponding factor above, the relation is seen to be the same.
The
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 ''D'', ''E'', and ''F'' are chosen on ''BC'', ''AC'', and ''AB'' respectively so that
:
then ''D'', ''E'', and ''F'' 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 ...
. The converse is often included as part of the theorem. (Note that the converse of the weaker, unsigned statement is not necessarily true.)
The theorem is very similar to
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 ...
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.
Proof
![Menelaos' theorem 2](https://upload.wikimedia.org/wikipedia/commons/4/41/Menelaos%27_theorem_2.svg)
A standard proof is as follows:
First, the sign of the
left-hand side
In mathematics, LHS is informal shorthand for the left-hand side of an equation. Similarly, RHS is the right-hand side. The two sides have the same value, expressed differently, since equality is symmetric.Pasch's axiom
In geometry, Pasch's axiom is a statement in plane geometry, used implicitly by Euclid, which cannot be derived from the postulates as Euclid gave them. Its essential role was discovered by Moritz Pasch in 1882.
Statement
The axiom states that, ...
.)
To check the magnitude, construct perpendiculars from ''A'', ''B'', and ''C'' to the line ''DEF'' and let their lengths be ''a, b,'' and ''c'' respectively. Then by
similar triangles it follows that , ''AF''/''FB'', = , ''a''/''b'', , , ''BD''/''DC'', = , ''b''/''c'', , and , ''CE''/''EA'', = , ''c''/''a'', . So
:
For a simpler, if less symmetrical way to check the magnitude, draw ''CK'' parallel to ''AB'' where ''DEF'' meets ''CK'' at ''K''. Then by similar triangles
:
and the result follows by eliminating ''CK'' from these equations.
The converse follows as a corollary. Let ''D'', ''E'', and ''F'' be given on the lines ''BC'', ''AC'', and ''AB'' so that the equation holds. Let ''F''′ be the point where ''DE'' crosses ''AB''. Then by the theorem, the equation also holds for ''D'', ''E'', and ''F''′. Comparing the two,
:
But at most one point can cut a segment in a given ratio so ''F''=''F''′.
A proof using homothecies
The following proof uses only notions 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 ...
, notably
homothecies.
Whether or not ''D'', ''E'', and ''F'' are collinear, there are three homothecies with centers ''D'', ''E'', ''F'' that respectively send ''B'' to ''C'', ''C'' to ''A'', and ''A'' to ''B''. The composition of the three then is an element of the group of homothecy-translations that fixes ''B'', so it is a homothecy with center ''B'', possibly with ratio 1 (in which case it is the identity). This composition fixes the line ''DE'' if and only if ''F'' is collinear with ''D'' and ''E'' (since the first two homothecies certainly fix ''DE'', and the third does so only if ''F'' lies on ''DE''). Therefore ''D'', ''E'', and ''F'' are collinear if and only if this composition is the identity, which means that the magnitude of product of the three ratios is 1:
:
which is equivalent to the given equation.
History
It is uncertain who actually discovered the theorem; however, the oldest extant exposition appears in ''Spherics'' by Menelaus. In this book, the plane version of the theorem is used as a lemma to prove a spherical version of the theorem.
In
Almagest
The ''Almagest'' is a 2nd-century Greek-language mathematical and astronomical treatise on the apparent motions of the stars and planetary paths, written by Claudius Ptolemy ( ). One of the most influential scientific texts in history, it canoni ...
,
Ptolemy
Claudius Ptolemy (; grc-gre, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific treatises, three of which were of importanc ...
applies the theorem on a number of problems in spherical astronomy.
During the
Islamic Golden Age
The Islamic Golden Age was a period of cultural, economic, and scientific flourishing in the history of Islam, traditionally dated from the 8th century to the 14th century. This period is traditionally understood to have begun during the reign ...
, Muslim scholars devoted a number of works that engaged in the study of Menelaus's theorem, which they referred to as "the proposition on the secants" (''shakl al-qatta). The
complete quadrilateral
In mathematics, specifically in incidence geometry and especially in projective geometry, a complete quadrangle is a system of geometric objects consisting of any four points in a plane, no three of which are on a common line, and of the six l ...
was called the "figure of secants" in their terminology.
Al-Biruni
Abu Rayhan Muhammad ibn Ahmad al-Biruni (973 – after 1050) commonly known as al-Biruni, was a Khwarazmian Iranian in scholar and polymath during the Islamic Golden Age. He has been called variously the "founder of Indology", "Father of Co ...
's work, ''The Keys of Astronomy'', lists a number of those works, which can be classified into studies as part of commentaries on Ptolemy's ''Almagest'' as in the works of
al-Nayrizi
Abū’l-‘Abbās al-Faḍl ibn Ḥātim al-Nairīzī ( ar, أبو العباس الفضل بن حاتم النيريزي, la, Anaritius, Nazirius, c. 865–922) was a Persian mathematician and astronomer from Nayriz, Fars Province, Iran.
He fl ...
and
al-Khazin
Abu Jafar Muhammad ibn Husayn Khazin ( fa, ابوجعفر خازن خراسانی; 900–971), also called Al-Khazin, was an Iranian Muslim astronomer and mathematician from Khorasan. He worked on both astronomy and number theory.
Al-Khazin was o ...
where each demonstrated particular cases of Menelaus's theorem that led to the
sine rule
In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law,
\frac \,=\, \frac \,=\, \frac \,=\, 2R,
where , and are ...
,
or works composed as independent treatises such as:
* The "Treatise on the Figure of Secants" (''Risala fi shakl al-qatta) by
Thabit ibn Qurra Thabit ( ar, ) is an Arabic name for males that means "the imperturbable one". It is sometimes spelled Thabet.
People with the patronymic
* Ibn Thabit, Libyan hip-hop musician
* Asim ibn Thabit, companion of Muhammad
* Hassan ibn Sabit (died 674 ...
.
*
Husam al-Din al-Salar's ''Removing the Veil from the Mysteries of the Figure of Secants'' (Kashf al-qina' 'an asrar al-shakl al-qatta'), also known as "The Book on the Figure of Secants" (''Kitab al-shakl al-qatta) or in Europe as ''The Treatise on the Complete Quadrilateral''. The lost treatise was referred to by
Sharaf al-Din al-Tusi
Sharaf al-Dīn al-Muẓaffar ibn Muḥammad ibn al-Muẓaffar al-Ṭūsī ( fa, شرفالدین مظفر بن محمد بن مظفر توسی; 1135 – 1213) was an Iranian mathematician and astronomer of the Islamic Golden Age (during the ...
and
Nasir al-Din al-Tusi
Muhammad ibn Muhammad ibn al-Hasan al-Tūsī ( fa, محمد ابن محمد ابن حسن طوسی 18 February 1201 – 26 June 1274), better known as Nasir al-Din al-Tusi ( fa, نصیر الدین طوسی, links=no; or simply Tusi in the West ...
.
* Work by
al-Sijzi
Abu Sa'id Ahmed ibn Mohammed ibn Abd al-Jalil al-Sijzi (c. 945 - c. 1020, also known as al-Sinjari and al-Sijazi; fa, ابوسعید سجزی; Al-Sijzi is short for "Al-Sijistani") was an Iranian Muslim astronomer, mathematician, and astrolog ...
.
* ''Tahdhib'' by
Abu Nasr ibn Iraq
Abu Nasri Mansur ibn Ali ibn Iraq ( fa, أبو نصر منصور بن علی بن عراق; c. 960 – 1036) was a Persian Muslim mathematician and astronomer. He is well known for his work with the spherical sine law.Bijli suggests that three m ...
.
*
Roshdi Rashed
Roshdi Rashed (Arabic: رشدي راشد), born in Cairo in 1936, is a mathematician, philosopher and historian of science, whose work focuses largely on mathematics and physics of the medieval Arab world. His work explores and illuminates the unr ...
and
Athanase Papadopoulos Athanase is a given name. Notable people with the name include:
*Antoine-Athanase Royer-Collard (1768–1825), French physician born in the village of Sompuis, département Marne
*Athanase Coquerel (disambiguation)
*Athanase David (1882–1953), C ...
, Menelaus' Spherics: Early Translation and al-Mahani'/al-Harawi's version (Critical edition of Menelaus' Spherics from the Arabic manuscripts, with historical and mathematical commentaries), De Gruyter, Series: Scientia Graeco-Arabica, 21, 2017, 890 pages.
References
*
External links
Alternate proofof Menelaus's theorem, from
PlanetMath
PlanetMath is a free, collaborative, mathematics online encyclopedia. The emphasis is on rigour, openness, pedagogy, real-time content, interlinked content, and also community of about 24,000 people with various maths interests. Intended to be c ...
Menelaus From CevaCeva and Menelaus Meet on the Roadsat MathPages
Demo of Menelaus's theoremby Jay Warendorff.
The 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 ...
.
*
{{Ancient Greek mathematics
Affine geometry
Theorems about triangles
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Euclidean plane geometry