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This article is about the theorem about the ratios of various line segments. For the special case of the inscribed angle theorem, see Thales's theorem.
The intercept theorem, also known as Thales's theorem, basic proportionality theorem or side splitter theorem is an important theorem in elementary geometry about the ratios of various line segments that are created if two rays with a common starting point are intercepted by a pair of parallels. It is equivalent đồ sộ the theorem about ratios in similar triangles. It is traditionally attributed đồ sộ Greek mathematician Thales. It was known đồ sộ the ancient Babylonians and Egyptians, although its first known proof appears in Euclid's Elements.
Formulation of the theorem
Suppose S is the common starting point of two rays and A, B are the intersections of the first ray with the two parallels, such that B is further away from S than vãn A, and similarly C, D are the intersections of the second ray with the two parallels such that D is further away from S than vãn C. In this configuration the following statements hold:
- The ratio of any two segments on the first ray equals the ratio of the according segments on the second ray:
- The ratio of the two segments on the same ray 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 rays 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.
Extensions and conclusions
The first two statements remain true if the two rays get replaced by two lines intersecting in . In this case there are two scenarios with regard đồ sộ , either it lies between the 2 parallels (X figure) or it does not (V figure). If is not located between the two parallels, the original theorem applies directly. If lies between the two paralles, then a reflection of and at yields V figure with identical measures for which the original theorem now applies. The third statement (converse) however does not remain true for lines.
If there are more than vãn two rays starting at or more than vãn two lines intersecting at , then each parallel contains more than vãn one line segment and the ratio of two line segments on one parallel equals the ratio of the according line segments on the other parallek. For instance if there's a third ray starting at and intersecting the parallels in and , such that is further away from than vãn , then the following equalities holds:
For the second equation the converse is true as well, that is if the 3 rays are intercepted two lines and the ratios of the according line segments on each line are equal, then those 2 lines must be parallel.
Similarity and similar triangles
The intercept theorem is closely related đồ sộ similarity. It is equivalent đồ sộ the concept of similar triangles, i.e. it can be used đồ sộ prove the properties of similar triangles and similar triangles can be used đồ sộ prove the intercept theorem. By matching identical angles you can always place two similar triangles in one another sánh 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, the axioms concerning the scalar multiplication (in particular and ) ensure that the intercept theorem holds. One has
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
- Doubling the cube
- Squaring the circle
It took more than vãn 2000 years until all three of them were finally shown đồ sộ be impossible with the given tools in the 19th century, using algebraic methods that had become available during that period of time. In order đồ sộ reformulate them in algebraic terms using field extensions, one needs đồ sộ match field operations with compass and straightedge constructions (see constructible number). In particular it is important đồ sộ 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 đồ sộ be able đồ sộ construct, for a line segment of length , a new line segment of length . The intercept theorem can be used đồ sộ show that in both cases such a construction is possible.
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Construction of a product
Construction of an inverse
Dividing a line segment in a given ratio
Measuring and survey
Height of the Cheops pyramid
According đồ sộ some historical sources the Greek mathematician Thales applied the intercept theorem đồ sộ determine the height of the Cheops' pyramid. The following mô tả tìm kiếm illustrates the use of the intercept theorem đồ sộ 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 đồ sộ apply the intercept theorem đồ sộ compute
Measuring the width of a river
Parallel lines in triangles and trapezoids
The intercept theorem can be used đồ sộ prove that a certain construction yields parallel line (segment)s.
If the midpoints of two triangle sides are connected then the resulting line segment is parallel đồ sộ the third triangle side (Midpoint theorem of triangles).
If the midpoints of the two non-parallel sides of a trapezoid are connected, then the resulting line segment is parallel đồ sộ the other two sides of the trapezoid.
The theorem is traditionally attributed đồ sộ the Greek mathematician Thales of Miletus, who may have used some khuông of the theorem đồ sộ determine heights of pyramids in Egypt and đồ sộ compute the distance of ship from the shore.    
An elementary proof of the theorem uses triangles of equal area đồ sộ derive the basic statements about the ratios (claim 1). The other claims then follow by applying the first claim and contradiction.
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- ^ a b Schupp, H. (1977). Elementargeometrie (in German). UTB Schöningh. pp. 124–126. ISBN 3-506-99189-2.
- ^ a b Strahlensätze. In: Schülerduden: Mathematik I. Dudenverlag, 8. edition, Mannheim 2008, pp. 431–433 (German)
- ^ Agricola, Ilka; Friedrich, Thomas (2008). Elementary Geometry. AMS. pp. 10–13, 16–18. ISBN 0-8218-4347-8. (online copy, p. 10, at Google Books)
- ^ a b c Lorenz Halbeisen, Norbert Hungerbühler, Juan Läuchli: Mit harmonischen Verhältnissen zu Kegelschnitten: Perlen der klassischen Geometrie. Springer năm nhâm thìn, ISBN 9783662530344, pp. 191–208 (German)
- ^ Kazarinoff, Nicholas D. (2003) , Ruler and the Round, Dover, p. 3, ISBN 0-486-42515-0
- ^ Kunz, Ernst (1991). Algebra (in German). Vieweg. pp. 5–7. ISBN 3-528-07243-1.
- ^ Ostermann, Alexander; Wanner, Gerhard (2012). Geometry by Its History. Springer. pp. 7. ISBN 978-3-642-29163-0. (online copy, p. 7, at Google Books)
- ^ No original work of Thales has survived. All historical sources that attribute the intercept theorem or related knowledge đồ sộ him were written centuries after his death. Diogenes Laertius and Pliny give a mô tả tìm kiếm 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 [Thales] 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 đồ sộ obtain the height of pyramids and all other similar objects, namely, by measuring the shadow of the object at the time when a toàn thân and its shadow are equal in length.". However, Plutarch gives an trương mục that may suggest Thales knowing the intercept theorem or at least a special case of it:".. without trouble or the assistance of any instrument [he] merely mix 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 đồ sộ the stick the same ratio which the shadow [of the pyramid] has đồ sộ the shadow [of 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 and Lives of Eminent Philosophers, Chapter 1. Thales, para.27)
- ^ Herbert Bruderer: Milestones in Analog and Digital Computing. Springer, 2021, ISBN 9783030409746, pp. 214–217
- ^ Dietmar Herrmann: Ancient Mathematics. History of Mathematics in Ancient Greece and Hellenism, Springer 2022, ISBN 978-3-662-66493-3, pp. 27-36
- ^ Francis Borceux: An Axiomatic Approach đồ sộ Geometry. Springer, 2013, pp. 10–13
- ^ Gilles Dowek: Computation, Proof, Machine. Cambridge University Press, năm ngoái, ISBN 9780521118019, pp. 17-18
- ^ Lothar Redlin, Ngo Viet, Saleem Watson: "Thales' Shadow", Mathematics Magazine, Vol. 73, No. 5 (Dec., 2000), pp. 347-353 (JSTOR
- French, Doug (2004). Teaching and Learning Geometry. BLoomsbury. pp. 84–87. ISBN 9780826473622. (online copy, p. 84, at Google Books)
- Agricola, Ilka; Friedrich, Thomas (2008). Elementary Geometry. AMS. pp. 10–13, 16–18. ISBN 0-8218-4347-8. (online copy, p. 10, at Google Books)
- Stillwell, John (2005). The Four Pillars of Geometry. Springer. p. 34. ISBN 978-0-387-25530-9. (online copy, p. 34, at Google Books)
- Ostermann, Alexander; Wanner, Gerhard (2012). Geometry by Its History. Springer. pp. 3–7. ISBN 978-3-642-29163-0. (online copy, p. 3, at Google Books)
- Lorenz Halbeisen, Norbert Hungerbühler, Juan Läuchli: Mit harmonischen Verhältnissen zu Kegelschnitten: Perlen der klassischen Geometrie. Springer năm nhâm thìn, ISBN 9783662530344, pp. 191–208 (German)
- Intercept Theorem at PlanetMath
- Alexander Bogomolny: Thales' Theorems and in particular Thales' Theorem at Cut-the-Knot
- intercept theorem interactive