Intercept theorem

The intercept theorem, also known as Thales's theorem or basic proportionality theorem, is an important theorem in elementary geometry about the ratios of various line segments 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. Traditionally it is attributed to Greek mathematician Thales.[1]

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.

1. The ratios of any two segments on the first line equals the ratios of the according segments on the second line: ${\displaystyle |SA|:|AB|=|SC|:|CD|}$, ${\displaystyle |SB|:|AB|=|SD|:|CD|}$, ${\displaystyle |SA|:|SB|=|SC|:|SD|}$
2. The ratio of the two segments on the same line starting at S equals the ratio of the segments on the parallels: ${\displaystyle |SA|:|SB|=|SC|:|SD|=|AC|:|BD|}$
3. The converse of the first statement is true as well, i.e. if the two intersecting lines are intercepted by two arbitrary lines and ${\displaystyle |SA|:|AB|=|SC|:|CD|}$ holds then the two intercepting lines are parallel. However the converse of the second statement is not true.
4. 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: ${\displaystyle |AF|:|BE|=|FC|:|ED|}$ , ${\displaystyle |AF|:|FC|=|BE|:|ED|}$

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

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 ${\displaystyle \lambda \cdot ({\vec {a}}+{\vec {b}})=\lambda \cdot {\vec {a}}+\lambda \cdot {\vec {b}}}$ and ${\displaystyle \|\lambda {\vec {a}}\|=|\lambda |\cdot \ \|{\vec {a}}\|}$) ensure that the intercept theorem holds. One has ${\displaystyle {\frac {\|\lambda \cdot {\vec {a}}\|}{\|{\vec {a}}\|}}={\frac {\|\lambda \cdot {\vec {b}}\|}{\|{\vec {b}}\|}}={\frac {\|\lambda \cdot ({\vec {a}}+{\vec {b}})\|}{\|{\vec {a}}+{\vec {b}}\|}}=|\lambda |}$

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:[2][3]

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 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 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 ${\displaystyle a}$, a new line segment of length ${\displaystyle a^{-1}}$. The intercept theorem can be used to show that in both cases 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 ${\displaystyle {\overline {AB}}}$ in a ${\displaystyle m:n}$ ratio, draw an arbitrary angle in A with ${\displaystyle {\overline {AB}}}$ as one leg. On the other leg construct ${\displaystyle m+n}$ equidistant points, then draw the line through the last point and B and parallel line through the mth point. This parallel line divides ${\displaystyle {\overline {AB}}}$ in the desired ratio. The graphic to the right shows the partition of a line segment ${\displaystyle {\overline {AB}}}$ in a ${\displaystyle 5:3}$ ratio.[4]

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.[1] 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

${\displaystyle C=65~{\text{m}}+{\frac {230~{\text{m}}}{2}}=180~{\text{m}}}$

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

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

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 ${\displaystyle |CF|}$,${\displaystyle |CA|}$,${\displaystyle |FE|}$ are measured and used to compute the wanted distance ${\displaystyle |AB|={\frac {|AC||FE|}{|FC|}}}$.

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.

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.[5]

Claim 1

Since ${\displaystyle CA\parallel BD}$, the altitudes of ${\displaystyle \triangle CDA}$ and ${\displaystyle \triangle CBA}$ are of equal length. As those triangles share the same baseline, their areas are identical. So we have ${\displaystyle |\triangle CDA|=|\triangle CBA|}$ and therefore ${\displaystyle |\triangle SCB|=|\triangle SDA|}$ as well. This yields ${\displaystyle {\frac {|\triangle SCA|}{|\triangle CDA|}}={\frac {|\triangle SCA|}{|\triangle CBA|}}}$ and ${\displaystyle {\frac {|\triangle SCA|}{|\triangle SDA|}}={\frac {|\triangle SCA|}{|\triangle SCB|}}}$ Plugging in the formula for triangle areas (${\displaystyle {\tfrac {{\text{baseline}}\cdot {\text{altitude}}}{2}}}$) transforms that into ${\displaystyle {\frac {|SC||AF|}{|CD||AF|}}={\frac {|SA||EC|}{|AB||EC|}}}$ and ${\displaystyle {\frac {|SC||AF|}{|SD||AF|}}={\frac {|SA||EC|}{|SB||EC|}}}$ Canceling the common factors results in: (a) ${\displaystyle \,{\frac {|SC|}{|CD|}}={\frac {|SA|}{|AB|}}}$ and (b) ${\displaystyle \,{\frac {|SC|}{|SD|}}={\frac {|SA|}{|SB|}}}$ Now use (b) to replace ${\displaystyle |SA|}$ and ${\displaystyle |SC|}$ in (a): ${\displaystyle {\frac {\frac {|SA||SD|}{|SB|}}{|CD|}}={\frac {\frac {|SB||SC|}{|SD|}}{|AB|}}}$ Using (b) again this simplifies to: (c) ${\displaystyle \,{\frac {|SD|}{|CD|}}={\frac {|SB|}{|AB|}}}$ ${\displaystyle \,\square }$

Claim 2

Draw an additional parallel to ${\displaystyle SD}$ through A. This parallel intersects ${\displaystyle BD}$ in G. Then one has ${\displaystyle |AC|=|DG|}$ and due to claim 1 ${\displaystyle {\frac {|SA|}{|SB|}}={\frac {|DG|}{|BD|}}}$ and therefore ${\displaystyle {\frac {|SA|}{|SB|}}={\frac {|AC|}{|BD|}}}$ ${\displaystyle \square }$

Claim 3

Assume ${\displaystyle AC}$ and ${\displaystyle BD}$ are not parallel. Then the parallel line to ${\displaystyle AC}$ through ${\displaystyle D}$ intersects ${\displaystyle SA}$ in ${\displaystyle B_{0}\neq B}$. Since ${\displaystyle |SB|:|SA|=|SD|:|SC|}$ is true, we have ${\displaystyle |SB|={\frac {|SD||SA|}{|SC|}}}$ and on the other hand from claim 1 we have ${\displaystyle |SB_{0}|={\frac {|SD||SA|}{|SC|}}}$. So ${\displaystyle B}$ and ${\displaystyle B_{0}}$ are on the same side of ${\displaystyle S}$ and have the same distance to ${\displaystyle S}$, which means ${\displaystyle B=B_{0}}$. This is a contradiction, so the assumption could not have been true, which means ${\displaystyle AC}$ and ${\displaystyle BD}$ are indeed parallel ${\displaystyle \square }$

Claim 4

Claim 4 can be shown by applying the intercept theorem for two lines.

Notes

1. 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)
2. Kazarinoff, Nicholas D. (2003) [1970], Ruler and the Round, Dover, p. 3, ISBN 0-486-42515-0
3. Kunz, Ernst (1991). Algebra (in German). Vieweg. pp. 5–7. ISBN 3-528-07243-1.
4. 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)
5. Schupp, H. (1977). Elementargeometrie (in German). UTB Schöningh. pp. 124–126. ISBN 3-506-99189-2.

References

• Schupp, H. (1977). Elementargeometrie (in German). UTB Schöningh. pp. 124–126. ISBN 3-506-99189-2.
• Leppig, Manfred (1981). Lernstufen Mathematik (in German). Girardet. pp. 157–170. ISBN 3-7736-2005-5.
• 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)

External links

• Intercept Theorem at PlanetMath
• Alexander Bogomolny: Thales' Theorems and in particular Thales' Theorem at Cut-the-Knot