Intercept theorem

File:Intercept theorem a.svg

File:Strahlensatz gegenbeispiel umkehrung 2ter satz.svg

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. The ratio of any two segments on the first ray equals the ratio of the according segments on the second ray:
   $\backslash frac$

   SA

   AB

   =\frac

   SC

   { | CD |}, $\backslash frac$

   SB

   AB

   =\frac

   SD

   CD

   , $\backslash frac$

   SA

   SB

   =\frac

   SC

   SD

2. The ratio of the two segments on the same ray starting at S equals the ratio of the segments on the parallels:
   $\backslash frac$

   SA

   SB

   = \frac

   SC

   SD

   =\frac

   AC

   BD

3. The converse of the first statement is true as well, i.e. if the two rays are intercepted by two arbitrary lines and $\backslash frac$

   SA

   AB

   =\frac

   SC

   CD

   holds then the two intercepting lines are parallel. However, the converse of the second statement is not true (see graphic for a counterexample).

File:Intercept theorem4.svg

File:Intercept theorem3.svg

The first two statements remain true if the two rays get replaced by two lines intersecting in $S$. In this case there are two scenarios with regard to $S$, either it lies between the 2 parallels (X figure) or it does not (V figure). If $S$ is not located between the two parallels, the original theorem applies directly. If $S$ lies between the two parallels, then a reflection of $A$ and $C$ at $S$ yields V figure with identical measures for which the original theorem now applies.

The third statement (converse) however does not remain true for lines instead of rays. However, if one replaces ratio of lengths with signed ratios of directed line segments, all statements of the intercept theorem including the converse remain valid for lines as well. More precisely if $A$ and $B$ are two points on a line and $C$ and $D$ are two points on the same line or on a parallel line, then the signed ratio $\backslash frac\{AB\}\{CD\}$ is $\backslash frac$

If there are more than two rays starting at $S$ or more than two lines intersecting at $S$, 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 $S$ and intersecting the parallels in $E$ and $F$, such that $F$ is further away from $S$ than $E$, then the following equalities hold:

: $\backslash frac$

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.

An homothety of positive ratio {{mvar|k}} with center {{mvar|O}} maps a point {{mvar|A}} to the point {{mvar|B}} located on the ray {{mvar|OA}} such that

:$\backslash frac$

The converse of the theorem implies that a homothety transforms a line in a parallel line.

Conversely, the direct statement of the intercept theorem implies that a geometric transformation is always a homothety of center {{mvar|O}}, if it fixes the lines passing through {{mvar|O}} and transforms every other line into a parallel line.

File:Intercept theorem- Triangles.svg

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.

In a normed vector space, the axioms concerning the scalar multiplication (in particular $\backslash lambda\; \backslash cdot\; (\backslash vec\{a\}+\backslash vec\{b\})=\backslash lambda\; \backslash cdot\; \backslash vec\{a\}+\; \backslash lambda\; \backslash cdot\; \backslash vec\{b\}$ and $\backslash |\backslash lambda\; \backslash vec\{a\}\backslash |=|\backslash lambda|\backslash cdot\backslash \; \backslash |\backslash vec\{a\}\backslash |$) ensure that the intercept theorem holds. One has

\frac{ \| \lambda \cdot \vec{a} \| }{ \| \vec{a} \|}

=\frac{\|\lambda\cdot\vec{b}\|}{\|\vec{b}\|}

=\frac{\|\lambda\cdot(\vec{a}+\vec{b}) \|}{\|\vec{a}+\vec{b}\|}

=|\lambda|

File:Intercept theorem vectors 2.svg

There are three famous problems in elementary geometry which were posed by the Greeks in terms of compass and straightedge constructions:{{citation|first=Nicholas D.|last=Kazarinoff|author-link=Nicholas D. Kazarinoff|title=Ruler and the Round|year=2003|orig-year=1970|publisher=Dover|page=3|isbn=0-486-42515-0}}

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 $a$, a new line segment of length $a^\{-1\}$. The intercept theorem can be used to show that for both cases, that such a construction is possible.

To divide an arbitrary line segment $\backslash overline\{AB\}$ in a $m:n$ ratio, draw an arbitrary angle in A with $\backslash overline\{AB\}$ as one leg. On the other leg construct $m+n$ equidistant points, then draw the line through the last point and B and parallel line through the mth point. This parallel line divides $\backslash overline\{AB\}$ in the desired ratio. The following graphic shows the partition of a line segment $\backslash overline\{AB\}$ in a $5:3$ ratio.

File:Dividing segment.svg

File:Thales Theorem 6.svg

File:Thales Theorem 7.svg

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.

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

:$C\; =\; 65~\backslash text\{m\}+\backslash frac\{230~\backslash text\{m\}\}\{2\}=180~\backslash text\{m\}$

Knowing A, B and C he was now able to apply the intercept theorem to compute

:$D=\backslash frac\{C\; \backslash cdot\; A\}\{B\}=\backslash frac\{1.63~\backslash text\{m\}\; \backslash cdot\; 180~\backslash text\{m\}\}\{2~\backslash text\{m\}\}=146.7~\backslash text\{m\}$

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 $|CF|$,$|CA|$,$|FE|$ are measured and used to compute the wanted distance $|AB|=\backslash frac$

File:River Chart.svg

The intercept theorem can be used to prove that a certain construction yields parallel line (segment)s.

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.

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.

Plugging in the formula for triangle areas ($\backslash tfrac\{\backslash text\{baseline\}\; \backslash cdot\; \backslash text\{altitude\}\}\{2\}$) transforms that into

$\backslash frac$

Canceling the common factors results in:

(a) $\backslash ,\; \backslash frac$

Now use (b) to replace $|SA|$ and $|SC|$ in (a):

$\backslash frac\{\backslash frac$

Using (b) again this simplifies to:

(c) $\backslash ,\; \backslash frac$

$\backslash ,\; \backslash square$

|-

|}

and therefore

$\backslash frac$

$\backslash square$

|-

|}

|-

|}

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: [http://www-history.mcs.st-and.ac.uk/Biographies/Thales.html Thales biography] of the MacTutor, the (translated) original works of Plutarch and Laertius are: [https://penelope.uchicago.edu/Thayer/E/Roman/Texts/Plutarch/Moralia/Dinner_of_the_Seven*.html#ref5 Moralia, The Dinner of the Seven Wise Men, 147A] and [https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0258%3Abook%3D1%3Achapter%3D1 Lives of Eminent Philosophers, Chapter 1. Thales, para.27])

{{Cite book |last=Schupp |first=H. |title=Elementargeometrie |publisher=UTB Schöningh |year=1977 |isbn=3-506-99189-2 |pages=124–126 |language=de}}

{{Cite book |last=Kunz |first=Ernst|title=Algebra|publisher=Vieweg |year=1991 |isbn=3-528-07243-1 |pages=5–7 |language=de}}

{{cite book |title=Geometry by Its History|url=https://archive.org/details/geometrybyitshis00oste|url-access=limited|first1=Alexander|last1=Ostermann|first2=Gerhard|last2=Wanner

|publisher=Springer|year=2012|isbn=978-3-642-29163-0|pages=[https://books.google.com/books?id=eOSqPHwWJX8C&pg=PA6 6–7]}}

Strahlensätze. In: Schülerduden: Mathematik I. Dudenverlag, 8. edition, Mannheim 2008, pp. 431–433 (German)

{{cite book |title=Elementary Geometry|first1=Ilka|last1=Agricola|author1-link= Ilka Agricola |first2=Thomas|last2=Friedrich

|publisher=AMS|year=2008|isbn=0-8218-4347-8|pages= 10–13, 16–18}} ({{Google books|LLXxBwAAQBAJ|online copy|page=10}})

Herbert Bruderer: Milestones in Analog and Digital Computing. Springer, 2021, {{ISBN|9783030409746}}, pp. 214–217

Lorenz Halbeisen, Norbert Hungerbühler, Juan Läuchli: Mit harmonischen Verhältnissen zu Kegelschnitten: Perlen der klassischen Geometrie. Springer 2016, {{ISBN|9783662530344}}, pp. 191–208 (German)

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 to Geometry. Springer, 2013, pp. 10–13

Gilles Dowek: Computation, Proof, Machine. Cambridge University Press, 2015, {{ISBN|9780521118019}}, pp. 17-18

Lothar Redlin, Ngo Viet, Saleem Watson: "Thales' Shadow", Mathematics Magazine, Vol. 73, No. 5 (Dec., 2000), pp. 347-353 ([https://www.jstor.org/stable/2690810 JSTOR])

See Agricola/Thomas or the following figure: {{pb}} File:Strahlensatz gegenbeispiel umkehrung geraden 1ter satz.svg

• {{cite book |title=Teaching and Learning Geometry|first1=Doug|last1=French|author1-link= Doug French|publisher=BLoomsbury|year=2004|isbn=9780826473622|pages= 84–87}} ({{Google books|j1nUAwAAQBAJ|online copy|page=84}})
• {{cite book |title=Elementary Geometry|first1=Ilka|last1=Agricola|author1-link= Ilka Agricola |first2=Thomas|last2=Friedrich

|publisher=AMS|year=2008|isbn=0-8218-4347-8|pages= 10–13, 16–18}} ({{Google books|LLXxBwAAQBAJ|online copy|page=10}})

• {{Cite book |first=John |last=Stillwell |title=The Four Pillars of Geometry |publisher=Springer |year=2005 |isbn=978-0-387-25530-9 |page=[https://archive.org/details/fourpillarsofgeo0000stil/page/34 34] |url=https://archive.org/details/fourpillarsofgeo0000stil/page/34 }} ({{Google books|fpAjJ6VJ3y8C|online copy|page=34}})
• {{cite book |title=Geometry by Its History|url=https://archive.org/details/geometrybyitshis00oste|url-access=limited|first1=Alexander|last1=Ostermann|first2=Gerhard|last2=Wanner

|publisher=Springer|year=2012|isbn=978-3-642-29163-0|pages=[https://archive.org/details/geometrybyitshis00oste/page/n16 3]–7}} ({{Google books|eOSqPHwWJX8C|online copy|page=3}})

• Lorenz Halbeisen, Norbert Hungerbühler, Juan Läuchli: Mit harmonischen Verhältnissen zu Kegelschnitten: Perlen der klassischen Geometrie. Springer 2016, {{ISBN|9783662530344}}, pp. 191–208 (German)

{{commons category}}

• [http://planetmath.org/intercepttheorem Intercept Theorem] at PlanetMath
• Alexander Bogomolny: [http://www.cut-the-knot.org/pythagoras/ThalesTheorems.shtml Thales' Theorems] and in particular [http://www.cut-the-knot.org/Curriculum/Geometry/GeoGebra/ThalesTheorem.shtml Thales' Theorem] at Cut-the-Knot
• [https://www.geogebra.org/m/p5ux2ybs intercept theorem interactive]

{{Ancient Greek mathematics}}

{{DEFAULTSORT:Intercept Theorem}}

Category:Euclidean geometry

Category:Theorems in plane geometry