Intercept 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 to the theorem about ratios in similar triangles. It is traditionally attributed to Greek mathematician Thales. It was known to the ancient Babylonians and Egyptians, although its first known proof appears in Euclid's Elements.

Formulation of the theorem

intercept theorem with rays
|SA| / |SB| = |AC| / |BD| does not necessarily imply AC is parallel to BD.

Suppose S is the common starting point of two rays, and two parallel lines are intersecting those two rays (see figure). Let A, B be the intersections of the first ray with the two parallels, such that B is further away from S than 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 C. In this configuration the following statements hold:[1][2]

  1. The ratio of any two segments on the first ray equals the ratio of the according segments on the second ray:
    , ,
  2. The ratio of the two segments on the same ray starting at S equals the ratio of the segments on the parallels:
  3. 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 (see graphic for a counterexample).

Extensions and conclusions

intercept theorem with a pair of intersecting lines
intercept theorem with more than two lines

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 to , 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 parallels, then a reflection of and at yields V figure with identical measures for which the original theorem now applies.[2] The third statement (converse) however does not remain true for lines.[3][4][5]

If there are more than two rays starting at or more than two lines intersecting at , then each parallel contains more than one line segment and the ratio of two line segments on one parallel equals the ratio of the according line segments on the other parallel. For instance if there's a third ray starting at and intersecting the parallels in and , such that is further away from than , then the following equalities holds:[4]

,

For the second equation the converse is true as well, that is if the 3 rays are intercepted by two lines and the ratios of the according line segments on each line are equal, then those 2 lines must be parallel.[4]

Similarity and similar triangles

Arranging two similar triangles, so that the intercept theorem can be applied

The intercept theorem is closely related to similarity. It is equivalent to the concept of similar triangles, 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, the axioms concerning the scalar multiplication (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:[6][7]

  1. Trisecting the angle
  2. Doubling the cube
  3. Squaring the circle

It took more than 2000 years until all three of them were finally shown to be impossible. This was achieved in the 19th century with the help of algebraic methods, that had become available by then. In order to reformulate the three problems in algebraic terms using field extensions, one needs to match field operations with compass and straightedge constructions (see constructible number). 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 for both cases, that such a construction is possible.

Construction of a product

Construction of an inverse

Dividing a line segment in a given ratio

To divide an arbitrary line segment in a ratio, draw an arbitrary angle in A with as one leg. On the other leg construct equidistant points, then draw the line through the last point and B and parallel line through the mth point. This parallel line divides in the desired ratio. The graphic to the right shows the partition of a line segment in a ratio.[8]

Measuring and survey

Height of the Cheops pyramid

measuring pieces
computing C and D

According to some historical sources the Greek mathematician Thales 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.[9][10]

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

The intercept theorem can be used to determine a distance that cannot be measured directly, such as the width of a river or a lake, the height of tall buildings or similar. The graphic to the right illustrates measuring the width of a river. The segments ,, are measured and used to compute the wanted distance .

Parallel lines in triangles and trapezoids

The intercept theorem can be used to 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 to 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 to the other two sides of the trapezoid.

Historical aspects

The theorem is traditionally attributed to the Greek mathematician Thales of Miletus, who may have used some form of the theorem to determine heights of pyramids in Egypt and to compute the distance of ship from the shore.[11][12][13][14]

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.[1]

Claim 1

Notation: For a triangle the vertical bars () denote its area and for a line segment its length.

Proof: Since , the altitudes of and are of equal length. As those triangles share the same baseline, their areas are identical. So we have and therefore as well. This yields

and

Plugging in the formula for triangle areas () transforms that into

and

Canceling the common factors results in:

(a) and (b)

Now use (b) to replace and in (a):

Using (b) again this simplifies to: (c)

Claim 2

Draw an additional parallel to through A. This parallel intersects in G. Then one has and due to claim 1 and therefore

Claim 3

Assume and are not parallel. Then the parallel line to through intersects in . Since is true, we have

and on the other hand from claim 1 we have
.
So and are on the same side of and have the same distance to , which means . This is a contradiction, so the assumption could not have been true, which means and are indeed parallel

Notes

  1. ^ a b Schupp, H. (1977). Elementargeometrie (in German). UTB Schöningh. pp. 124–126. ISBN 3-506-99189-2.
  2. ^ a b Strahlensätze. In: Schülerduden: Mathematik I. Dudenverlag, 8. edition, Mannheim 2008, pp. 431–433 (German)
  3. ^ 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)
  4. ^ a b c Lorenz Halbeisen, Norbert Hungerbühler, Juan Läuchli: Mit harmonischen Verhältnissen zu Kegelschnitten: Perlen der klassischen Geometrie. Springer 2016, ISBN 9783662530344, pp. 191–208 (German)
  5. ^ See Agricola/Thomas or the following figure:
    |SA| / |SB| = |SC| / |SD| does not necessarily imply AC is parallel to BD.
  6. ^ Kazarinoff, Nicholas D. (2003) [1970], Ruler and the Round, Dover, p. 3, ISBN 0-486-42515-0
  7. ^ Kunz, Ernst (1991). Algebra (in German). Vieweg. pp. 5–7. ISBN 3-528-07243-1.
  8. ^ 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)
  9. ^ 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 and Pliny 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 [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 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 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 [he] merely 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 [of the pyramid] has to 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)
  10. ^ Herbert Bruderer: Milestones in Analog and Digital Computing. Springer, 2021, ISBN 9783030409746, pp. 214–217
  11. ^ Dietmar Herrmann: Ancient Mathematics. History of Mathematics in Ancient Greece and Hellenism, Springer 2022, ISBN 978-3-662-66493-3, pp. 27-36
  12. ^ Francis Borceux: An Axiomatic Approach to Geometry. Springer, 2013, pp. 10–13
  13. ^ Gilles Dowek: Computation, Proof, Machine. Cambridge University Press, 2015, ISBN 9780521118019, pp. 17-18
  14. ^ Lothar Redlin, Ngo Viet, Saleem Watson: "Thales' Shadow", Mathematics Magazine, Vol. 73, No. 5 (Dec., 2000), pp. 347-353 (JSTOR

References

Read other articles:

This is the talk page for discussing improvements to the WikiProject Organized Labour page. Put new text under old text. Click here to start a new topic. New to Wikipedia? Welcome! Learn to edit; get help. Assume good faith Be polite and avoid personal attacks Be welcoming to newcomers Seek dispute resolution if needed Archives: 1, 2, 3, 4, 5, 6, 7, 8, 9Auto-archiving period: 30 days  Organized Labour Project‑class Organized Labour portalThis page is within the scope of WikiProject Organ...

 

1971 Asian Champion Club TournamentMaccabi Tel Aviv players with the trophyTournament detailsHost countryThailandDates21 March – 2 April 1971Teams8Venue(s)BangkokFinal positionsChampions Maccabi Tel Aviv (2nd title)Runners-up Aliyat Al-ShortaThird place Taj TehranFourth place ROK ArmyTournament statisticsTop scorer(s) Sabah Hatem Shlomo Gerbi Ali Al-Mulla(4 goals each)Best goalkeeper Sattar Khalaf← 1970 1972 → International football competition The 1971 Asian Champion Club Tour...

 

19th-century American temperance activist and a founder of the Prohibition Party James Black3rd Chairman of the Prohibition PartyIn office1876–1880Preceded bySimeon B. ChaseSucceeded byGideon T. Stewart Personal detailsBorn(1823-07-23)July 23, 1823Lewisburg, Pennsylvania, U.S.DiedDecember 16, 1893(1893-12-16) (aged 70)Lancaster, Pennsylvania, U.S.Political partyProhibitionOther politicalaffiliationsDemocratic Party (before 1854)Republican Party (1854-1869)SpouseEliza MurrayChildren6Par...

December 2013 North American storm complexSatellite image from NASA depicting the system over the Central United States on 21 December. TypeIce stormWinter stormTornado outbreakExtratropical cycloneFormed19 December 2013Dissipated23 December 2013 Lowest pressure997 mb (29.44 inHg) Tornadoesconfirmed13Max. rating1EF2 tornadoDuration oftornado outbreak22 days, 6 hours and 4 minutes Maximum snowfallor ice accretionSnowfall – ~36 cm (14 in)Ice – Around 30 mm (1.2...

 

La ragazza con la pistolaMonica Vitti in una scena del filmPaese di produzioneItalia, Regno Unito Anno1968 Durata102 min Rapporto1,85:1 Generecommedia RegiaMario Monicelli SoggettoRodolfo Sonego SceneggiaturaRodolfo Sonego, Luigi Magni ProduttoreGianni Hecht Lucari Produttore esecutivoFausto Saraceni Distribuzione in italianoEuro International Films FotografiaCarlo Di Palma MontaggioRuggero Mastroianni MusichePeppino De Luca CostumiMaurizio Chiari Interpreti e personaggi Monica Vitti: Ass...

 

Este artículo trata sobre la revista de televisión. Para horarios en plantilla sobre programación televisiva en general, véase Guía electrónica de programas. TV Guide Logotipo para TV Guide.Jefe de redacción Debra BirnbaumCategorías Noticias y entretenimientoFrecuencia SemanalCirculación 2 400 000 000 copiasIntroducida 3 de abril de 1953 (71 años)Sede Radnor Township, PensilvaniaIdioma InglésSitio web http://www.tvguide.comISSN 0039-8543[editar datos en Wik...

Artikel ini sebatang kara, artinya tidak ada artikel lain yang memiliki pranala balik ke halaman ini.Bantulah menambah pranala ke artikel ini dari artikel yang berhubungan atau coba peralatan pencari pranala.Tag ini diberikan pada Maret 2009. Kartu Pos ini memperlihatkan sebuah kereta kuda melewati Gedung Harmoni yang masih berdiri di di sudut Rijswijk (kini Jalan Veteran) dan Rijswijkstraat (kini Jl. Majapahit). Gedung Harmoni (Belanda: Societeit Harmonie) adalah gedung Belanda yang dulu ter...

 

Kota AlexandriaCity of AlexandriaGeorge Washington Masonic National Memorial pada tahun 2015 dengan Washington, D.C., dan Arlington dikejauhan BenderaLambangNegara Amerika SerikatNegara bagian VirginiaDidirikan1749Tergabung1870Pemerintahan • Wali kotaWilliam D. Euille (D)Luas • Total39,9 km2 (15,4 sq mi) • Luas daratan39,3 km2 (15,2 sq mi) • Luas perairan0,6 km2 (0,2 sq mi)Ketinggian12 m (39...

 

Генуциилат. Genuciiлат. gens Genucia Ветви рода Авгурин, Авентинен, Клепсина Подданство Древний Рим Гражданская деятельность консулы, военный трибун с консульской властью, децимвир, народные трибуны Военная деятельность полководцы, военные трибуны Религиозная деятельност...

6th edition of CHAN 2020 African Nations ChampionshipChampionnat d'Afrique des Nations 20202020 CHAN / CHAN 2020Tournament detailsHost countryCameroonDates16 January – 7 February 2021Teams16 (from 1 confederation)Venue(s)4 (in 3 host cities)Final positionsChampions Morocco (2nd title)Runners-up MaliThird place GuineaFourth place CameroonTournament statisticsMatches played32Goals scored62 (1.94 per match)Top scorer(s) Soufiane Rahimi (5 goals)Best pla...

 

Эта статья — о протестах марта 2023 года. О протестах 2024 года см. Протесты в Грузии против закона об «иноагентах» (2024). Протесты в Грузии (2023) Протестующие и полиция в Тбилиси Дата 6—10 марта 2023 Место Грузия Причины Принятый в первом чтении закон об иностранных аген�...

 

Atraktor Lorenz adalah contoh sistem dinamik nonlinear. Studi terhadap sistem ini membantu munculnya teori khaos. Teori sistem dinamik adalah bidang matematika terapan yang digunakan untuk memerikan kelakuan sistem dinamik kompleks, biasanya dengan menggunakan persamaan diferensial ataupun persamaan beda. Bila digunakan persamaan diferensial, teori tersebut dinamakan sistem dinamik kontinu. Bila digunakan persamaan beda, teori tersebut dinamakan sistem dinamik diskret. Bila variabel waktu ber...

Навчально-науковий інститут інноваційних освітніх технологій Західноукраїнського національного університету Герб навчально-наукового інституту інноваційних освітніх технологій ЗУНУ Скорочена назва ННІІОТ ЗУНУ Основні дані Засновано 2013 Заклад Західноукраїнський �...

 

Golf practice facility For driving range for vehicles, see electric vehicle battery. Driving range with 43 tees (20 covered) at the University of Washington Two-story driving range in Kanagawa, Japan (June 2023) A driving range is a facility or area where golfers can practice their golf swing. It can also be a recreational activity itself for amateur golfers or when enough time for a full game is not available. Many golf courses have a driving range attached and they are also found as stand-a...

 

Daily newspaper published in Bismarck, North Dakota, U.S. This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: The Bismarck Tribune – news · newspapers · books · scholar · JSTOR (July 2017) (Learn how and when to remove this message) The Bismarck TribuneThe March 1, 2012 front page of The Bismarck TribuneTypeDail...

Wakil Bupati PosoLambang Kabupaten PosoPetahanaYasin Mangunsejak 26 Februari 2021KediamanRumah Jabatan Wakil Bupati PosoMasa jabatan5 tahunDibentuk1994Pejabat pertamaAbdul Malik SyahadatSitus webhttp://posokab.go.id/ Wakil Bupati Poso adalah seseorang yang memegang kekuasaan tertinggi kedua setelah Bupati dalam lingkup Pemerintahan Kabupaten Poso di Poso, Sulawesi Tengah, Indonesia. Pada dasarnya, Wakil Bupati Poso memiliki tugas dan wewenang mendampingi dan mewakili Bupati dalam memimpi...

 

UnivisionJenisJaringan televisi terestrialNegara United StatesKantor pusat: New York CityProduction: Doral, FloridaJangkauanthroughout the United StatesSlogan • Todos estamos con Univisión(We all are with Univision) • Estás en Casa (You are at Home; used in Puerto Rico) • Siempre Contigo (Always With You) • La Casa de Todos (The House of All)Wilayah siar United States, Puerto RicoPemilikUnivision Communications, Inc. (95%) and Grupo Televisa (5%)To...

 

Election in Texas Main article: 1892 United States presidential election 1892 United States presidential election in Texas ← 1888 November 8, 1892 1896 →   Nominee Grover Cleveland James B. Weaver Benjamin Harrison Party Democratic Populist Republican Home state New York Iowa Indiana Running mate Adlai Stevenson I James G. Field Whitelaw Reid Electoral vote 15 0 0 Popular vote 239,148 99,688 81,144 Percentage 56.65% 23.61% 19.22% County Results C...

يفتقر محتوى هذه المقالة إلى الاستشهاد بمصادر. فضلاً، ساهم في تطوير هذه المقالة من خلال إضافة مصادر موثوق بها. أي معلومات غير موثقة يمكن التشكيك بها وإزالتها. (يوليو 2016) منتخب الاتحاد السوفيتي تحت 21 سنة لكرة القدم الفئة كرة قدم تحت 21 سنة للرجال  [لغات أخرى]‏  رمز الف�...

 

Komisi Kesehatan Nasional Republik Rakyat Tiongkok中华人民共和国国家卫生健康委员会Zhōnghuá Rénmín Gònghéguó Guójiā Wèishēng Jiànkāng WěiyuánhuìLambang Republik Rakyat TiongkokInformasi lembagaDibentuk19 Maret 2018 (2018-03-19)Nomenklatur lembaga sebelumnyaKomisi Kesehatan dan Keluarga Berencana NasionalWilayah hukumRepublik Rakyat TiongkokKantor pusatBeijingMenteriMa Xiaowei[1], Menteri bertanggung jawab atas Komisi Kesehatan Nasional Republik Raky...