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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 segment In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between i ...
s that are created if two intersecting lines 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 wi ...
. 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: , SA , : , AB , =, SC , : , CD , , , SB , : , AB , =, SD , : , CD , , , SA , : , SB , =, SC , : , SD , # The ratio of the two segments on the same line starting at S equals the ratio of the segments on the parallels: , SA , :, SB , = , SC , :, SD , =, AC , : , BD , #The converse of the first statement is true as well, i.e. if the two intersecting lines are intercepted by two arbitrary lines and , SA , : , AB , =, SC , : , CD , 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: , AF , : , BE , =, FC , : , ED , , , AF , : , FC , =, BE , : , ED , ::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 wi ...
, 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 In logic and mathematics, the converse of a categorical or implicational statement is the result of reversing its two constituent statements. For the implication ''P'' → ''Q'', the converse is ''Q'' → ''P''. For the categorical proposit ...
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
axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...
s 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 \lambda \cdot (\vec+\vec)=\lambda \cdot \vec+ \lambda \cdot \vec and \, \lambda \vec\, =, \lambda, \cdot\ \, \vec\, ) ensure that the intercept theorem holds. One has \frac =\frac =\frac =, \lambda,


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 In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an ideali ...
: #
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 an ...
#
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 Squaring the circle is a problem in geometry first proposed in Greek mathematics. It is the challenge of constructing a square with the area of a circle by using only a finite number of steps with a compass and straightedge. The difficul ...
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 extension In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
s, 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 , a new line segment of length a^ . 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 : C = 65~\text+\frac=180~\text Knowing A,B and C he was now able to apply the intercept theorem to compute : D=\frac=\frac=146.7~\text


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 (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 pyramidhas to the shadow f the stick'". (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 Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...

intercept theorem interactive
{{DEFAULTSORT:Intercept Theorem Euclidean geometry Theorems in plane geometry