Difference between revisions of "Menelaus theorem"
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| − | A theorem on the relations between the lengths of the segments on the sides of a triangle determined by an intersecting straight line. It asserts that if the given line intersects the sides of a triangle | + | {{TEX|done}} |
| + | A theorem on the relations between the lengths of the segments on the sides of a triangle determined by an intersecting straight line. It asserts that if the given line intersects the sides of a triangle $ABC$ (or their extensions) at the points $C'$, $A'$ and $B'$, then | ||
| − | + | $$\frac{AC'}{BC'}\cdot\frac{BA'}{CA'}\cdot\frac{CB'}{AB'}=1.$$ | |
| − | Menelaus' theorem is a particular case of the [[Carnot theorem|Carnot theorem]]; it can be generalized to the case of a [[Polygon|polygon]]. Thus, suppose that a straight line | + | Menelaus' theorem is a particular case of the [[Carnot theorem|Carnot theorem]]; it can be generalized to the case of a [[Polygon|polygon]]. Thus, suppose that a straight line $l$ intersects the edges $A_1A_2,\dots,A_{n-1}A_n,A_nA_1$ of a polygon $A_1\dots A_n$ at the respective points $a_1,\dots,a_n$. Then the following relation is valid: |
| − | + | $$\frac{A_1a_1}{A_2a_1}\cdots\frac{A_{n-1}a_{n-1}}{A_na_{n-1}}\cdot\frac{A_na_n}{A_na_n}=1.$$ | |
The theorem was proved by Menelaus (1st century) and apparently it was known to Euclid (3rd century B.C.). | The theorem was proved by Menelaus (1st century) and apparently it was known to Euclid (3rd century B.C.). | ||
Latest revision as of 14:39, 19 August 2014
A theorem on the relations between the lengths of the segments on the sides of a triangle determined by an intersecting straight line. It asserts that if the given line intersects the sides of a triangle $ABC$ (or their extensions) at the points $C'$, $A'$ and $B'$, then
$$\frac{AC'}{BC'}\cdot\frac{BA'}{CA'}\cdot\frac{CB'}{AB'}=1.$$
Menelaus' theorem is a particular case of the Carnot theorem; it can be generalized to the case of a polygon. Thus, suppose that a straight line $l$ intersects the edges $A_1A_2,\dots,A_{n-1}A_n,A_nA_1$ of a polygon $A_1\dots A_n$ at the respective points $a_1,\dots,a_n$. Then the following relation is valid:
$$\frac{A_1a_1}{A_2a_1}\cdots\frac{A_{n-1}a_{n-1}}{A_na_{n-1}}\cdot\frac{A_na_n}{A_na_n}=1.$$
The theorem was proved by Menelaus (1st century) and apparently it was known to Euclid (3rd century B.C.).
Figure: m063400a
Comments
References
| [a1] | B.L. van der Waerden, "Science awakening" , 1 , Noordhoff & Oxford Univ. Press (1961) pp. 275 (Translated from Dutch) |
How to Cite This Entry:
Menelaus theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Menelaus_theorem&oldid=19060
Menelaus theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Menelaus_theorem&oldid=19060
This article was adapted from an original article by P.S. Modenov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article