Intercept theorem

The intercept theorem is an important theorem in elementary geometry about the ratios of various line segments, that are created if 2 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, which is the reason why it is named theorem of Thales in some languages.

Formulation

S is the point the intersection of 2 lines and A,B are the intersections of the first line with the 2 parallels, such that B is further away from S than A, and similarly are C, D the intersections of the second line with the 2 parallels such that D is further away from S than C.

# The ratios of the any 2 segments on the first line equals the rations of the according segments on the second line: $|\; SA\; |\; :\; |\; AB\; |\; =|\; SC\; |\; :\; |\; CD\; |$, $|\; SB\; |\; :\; |\; AB\; |\; =|\; SD\; |\; :\; |\; CD\; |$, $|\; SA\; |\; :\; |\; SB\; |\; =|\; SC\; |\; :\; |\; SD\; |$
# The ratio of the 2 segments on the same line starting at S equals the ratio of the segments on the parallels: $|\; SA\; |:|\; SB\; |\; =\; |\; SC\; |\; :|\; SD\; |\; =|\; AC\; |\; :\; |\; BD\; |$
# The converse of the first statement is true as well, i.e. if the 2 intersecting lines are intercepted by 2 arbitrary lines and $|\; SA\; |\; :\; |\; AB\; |\; =|\; SC\; |\; :\; |\; CD\; |$ holds then the 2 intercepting lines are parallel. However the converse of the second statement is not true.
# If you have more than 2 lines intersecting in S, then ratio of the 2 segments on a parallel equals the ratio of the according segments on the other parallel. An example for the case of 3 lines is given the second graphic below.

Related Concepts

imilarity and similar Triangles

The intercept theorem is closely related to the similarity. In fact 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 2 similar triangles in one another, so that you get the configuration in which the intercepts applies and vice versa the intercept theorem configuartion contains always 2 similar triangles.

Arranging 2 similar triangles, so that the intercept theorem can be applied

Measuring/Survey

Height of the Cheops Pyramid

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Parallel Lines in Triangles and Trapezoids

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

=frac=frac=frac=frac=frac=frac$,\; square$

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claim 2

=frac=frac

claim 3

and on the other hand from claim 2 we have $|SB\_\{0\}|=frac$

claim 4

Can be shown by applying the intercept theorem for 2 lines.

ee also

*Similarity
*Thales

External links

*http://www.mathsrevision.net/gcse/pages.php?page=28
*http://kilian.ifastnet.com/applets_co/intercept_theorem/intercept_theorem.html
*http://www-history.mcs.st-and.ac.uk/Biographies/Thales.html

References

*Schupp, H.: "Elementargeometrie". UTB Schöningh 1977, ISBN 3-506-99189-2