 Intercept theorem
The intercept theorem is an important theorem in
elementary geometry about the ratios of variousline segment s, that are created if 2 intersectingline s are intercepted by a pair ofparallel s. It is equivalent to the theorem about ratios insimilar triangles . Traditionally it is attributed to Greek mathematicianThales , 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 ofsimilar 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
.

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$,\; square$

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://wwwhistory.mcs.stand.ac.uk/Biographies/Thales.htmlReferences
*Schupp, H.: "Elementargeometrie". UTB Schöningh 1977, ISBN 3506991892
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