The intercept theorem, also known as Thales's theorem, basic proportionality theorem or side splitter theorem is an important theorem in
elementary 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 ca ...
about the ratios of various
line segments that are created if two intersecting
lines
Line most often refers to:
* Line (geometry), object with zero thickness and curvature that stretches to infinity
* Telephone line, a single-user circuit on a telephone communication system
Line, lines, The Line, or LINE may also refer to:
Arts ...
are intercepted by a pair of
parallels. It is equivalent to the theorem about ratios in
similar triangles
In Euclidean geometry, two objects are similar if they have the same shape, or one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling (enlarging or reducing), possibly wit ...
. It is traditionally attributed to Greek mathematician
Thales
Thales of Miletus ( ; grc-gre, Θαλῆς; ) was a Greek mathematician, astronomer, statesman, and pre-Socratic philosopher from Miletus in Ionia, Asia Minor. He was one of the Seven Sages of Greece. Many, most notably Aristotle, regarded ...
.
It was known to the ancient Babylonians and Egyptians, although its first known proof appears in
Euclid
Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
's ''
Elements''.
Formulation
Suppose S is the intersection point of two lines and A, B are the intersections of the first line with the two parallels, such that B is further away from S than A, and similarly C, D are the intersections of the second line with the two parallels such that D is further away from S than C.
# The ratio of any two segments on the first line equals the ratio of the according segments on the second line:
,
,
# The ratio of the two segments on the same line starting at S equals the ratio of the segments on the parallels:
#The converse of the first statement is true as well, i.e. if the two intersecting lines are intercepted by two arbitrary lines and
holds then the two intercepting lines are parallel. However, the converse of the second statement is not true.
# If you have more than two lines intersecting in S, then ratio of the two segments on a parallel equals the ratio of the according segments on the other parallel:
,
::An example for the case of three lines is given in the second graphic below.
The first intercept theorem shows the ratios of the sections from the lines, the second the ratios of the sections from the lines as well as the sections from the parallels, finally the third shows the ratios of the sections from the parallels.
Related concepts
Similarity and similar triangles
The intercept theorem is closely related to
similarity. It is equivalent to the concept of
similar triangles
In Euclidean geometry, two objects are similar if they have the same shape, or one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling (enlarging or reducing), possibly wit ...
, i.e. it can be used to prove the properties of similar triangles and similar triangles can be used to prove the intercept theorem. By matching identical angles you can always place two similar triangles in one another so that you get the configuration in which the intercept theorem applies; and
conversely the intercept theorem configuration always contains two similar triangles.
Scalar multiplication in vector spaces
In a normed
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
, the
axioms concerning the
scalar multiplication
In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In common geometrical contexts, scalar multiplication of a real Euclidean vector b ...
(in particular
and
) ensure that the intercept theorem holds. One has
Applications
Algebraic formulation of compass and ruler constructions
There are three famous problems in elementary geometry which were posed by the Greeks in terms of
compass and straightedge constructions:
[
# ]Trisecting the angle
Angle trisection is a classical problem of straightedge and compass construction of ancient Greek mathematics. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge a ...
# Doubling the cube
Doubling the cube, also known as the Delian problem, is an ancient geometric problem. Given the edge of a cube, the problem requires the construction of the edge of a second cube whose volume is double that of the first. As with the related probl ...
# Squaring the circle
It took more than 2000 years until all three of them were finally shown to be impossible with the given tools in the 19th century, using algebraic methods that had become available during that period of time.
In order to reformulate them in algebraic terms using field extensions, one needs to match field operations with compass and straightedge constructions (see constructible number
In geometry and algebra, a real number r is constructible if and only if, given a line segment of unit length, a line segment of length , r, can be constructed with compass and straightedge in a finite number of steps. Equivalently, r is cons ...
). In particular it is important to assure that for two given line segments, a new line segment can be constructed such that its length equals the product of lengths of the other two. Similarly one needs to be able to construct, for a line segment of length , a new line segment of length . The intercept theorem can be used to show that in both cases such a construction is possible.
Dividing a line segment in a given ratio
Measuring and survey
Height of the Cheops pyramid
According to some historical sources the Greek mathematician Thales
Thales of Miletus ( ; grc-gre, Θαλῆς; ) was a Greek mathematician, astronomer, statesman, and pre-Socratic philosopher from Miletus in Ionia, Asia Minor. He was one of the Seven Sages of Greece. Many, most notably Aristotle, regarded ...
applied the intercept theorem to determine the height of the Cheops' pyramid. The following description illustrates the use of the intercept theorem to compute the height of the pyramid. It does not, however, recount Thales' original work, which was lost.
Thales measured the length of the pyramid's base and the height of his pole. Then at the same time of the day he measured the length of the pyramid's shadow and the length of the pole's shadow. This yielded the following data:
* height of the pole (A): 1.63 m
* shadow of the pole (B): 2 m
* length of the pyramid base: 230 m
* shadow of the pyramid: 65 m
From this he computed
:
Knowing A,B and C he was now able to apply the intercept theorem to compute
:
Measuring the width of a river
Parallel lines in triangles and trapezoids
The intercept theorem can be used to prove that a certain construction yields parallel line (segment)s.
Proof
An elementary proof of the theorem uses triangles of equal area to derive the basic statements about the ratios (claim 1). The other claims then follow by applying the first claim and contradiction.[
]
Claim 1
Claim 2
Claim 3
Claim 4
Claim 4 can be shown by applying the intercept theorem for two lines.
Notes
[No original work of Thales has survived. All historical sources that attribute the intercept theorem or related knowledge to him were written centuries after his death. ]Diogenes Laertius
Diogenes Laërtius ( ; grc-gre, Διογένης Λαέρτιος, ; ) was a biographer of the Greek philosophers. Nothing is definitively known about his life, but his surviving ''Lives and Opinions of Eminent Philosophers'' is a principal sour ...
and Pliny
Pliny may refer to:
People
* Pliny the Elder (23–79 CE), ancient Roman nobleman, scientist, historian, and author of ''Naturalis Historia'' (''Pliny's Natural History'')
* Pliny the Younger (died 113), ancient Roman statesman, orator, w ...
give a description that strictly speaking does not require the intercept theorem, but can rely on a simple observation only, namely that at a certain point of the day the length of an object's shadow will match its height. Laertius quotes a statement of the philosopher Hieronymus
Hieronymus, in English pronounced or , is the Latin form of the Ancient Greek name (Hierṓnymos), meaning "with a sacred name". It corresponds to the English given name Jerome.
Variants
* Albanian: Jeronimi
* Arabic: جيروم (Jerome)
* Basq ...
(3rd century BC) about Thales: "''Hieronymus says that hales
Hales is a small village in Norfolk, England. It covers an area of and had a population of 479 in 192 households as of the 2001 census, which had reduced to 469 at the 2011 census.
History
The villages name means 'Nooks of land'.
The manor ...
measured the height of the pyramids by the shadow they cast, taking the observation at the hour when our shadow is of the same length as ourselves (i.e. as our own height).''". Pliny writes: "''Thales discovered how to obtain the height of pyramids and all other similar objects, namely, by measuring the shadow of the object at the time when a body and its shadow are equal in length.''". However, Plutarch
Plutarch (; grc-gre, Πλούταρχος, ''Ploútarchos''; ; – after AD 119) was a Greek Middle Platonist philosopher, historian, biographer, essayist, and priest at the Temple of Apollo in Delphi. He is known primarily for hi ...
gives an account that may suggest Thales knowing the intercept theorem or at least a special case of it:"''.. without trouble or the assistance of any instrument emerely set up a stick at the extremity of the shadow cast by the pyramid and, having thus made two triangles by the intercept of the sun's rays, ... showed that the pyramid has to the stick the same ratio which the shadow f the pyramid
F, or f, is the sixth Letter (alphabet), letter in the Latin alphabet, used in the English alphabet, modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is English alphabet#Let ...
has to the shadow f the stick
F, or f, is the sixth letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''ef'' (pronounced ), and the plural is ''efs''.
Hist ...
'". (Source
''Thales biography''
of the MacTutor, the (translated) original works of Plutarch and Laertius are
''Moralia, The Dinner of the Seven Wise Men'', 147A
an
''Lives of Eminent Philosophers'', Chapter 1. Thales, para.27
*[ ()]
References
* ()
* ()
* ()
* ()
External links
''Intercept Theorem''
at 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 ...
*Alexander Bogomolny
''Thales' Theorems''
and in particula
''Thales' Theorem''
at Cut-the-Knot
intercept theorem interactive
{{DEFAULTSORT:Intercept Theorem
Euclidean geometry
Theorems in plane geometry