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Basic Proportionality Theorem
On ratios of line segments formed when 2 intersecting lines are cut by a pair of parallels
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.
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 ratios of any two segments on the first line equals the ratios of the according segments on the second line: , ,
The ratio of the two segments on the same line 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 intersecting lines 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.
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: ,
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.
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.
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 , a new line segment of length . 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 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.
Measuring and survey
Height of the Cheops pyramid
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.
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.
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.
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
Plugging in the formula for triangle areas () transforms that into
Canceling the common factors results in:
(a) and (b)
Now use (b) to replace and in (a):
Using (b) again this simplifies to:
Draw an additional parallel to through A. This parallel intersects in G. Then one has and due to claim 1
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
Claim 4 can be shown by applying the intercept theorem for two lines.
^ abNo 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)
^Kazarinoff, Nicholas D. (2003) , Ruler and the Round, Dover, p. 3, ISBN0-486-42515-0