Difference between revisions of "Triangle Inequality"
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| − | The Triangle Inequality can also be extended to other [[polygon]]s. The lengths <math>a_1, a_2, \ldots, a_n</math> can only be the sides of a nondegenerate <math>n</math>-gon if <math>a_i < a_1 + \ldots + a_{i -1} + a_{i + 1} + \ldots + a_n = \left(\sum_{j=1}^n a_j\right) - a_i</math> for <math>i = 1, 2 \ldots, n</math>. Expressing the inequality in this form leads to <math>2a_i < P</math>, where <math>P</math> is the sum of the <math>a_j</math>, or <math> | + | The Triangle Inequality can also be extended to other [[polygon]]s. The lengths <math>a_1, a_2, \ldots, a_n</math> can only be the sides of a nondegenerate <math>n</math>-gon if <math>a_i < a_1 + \ldots + a_{i -1} + a_{i + 1} + \ldots + a_n = \left(\sum_{j=1}^n a_j\right) - a_i</math> for <math>i = 1, 2 \ldots, n</math>. Expressing the inequality in this form leads to <math>2a_i < P</math>, where <math>P</math> is the sum of the <math>a_j</math>, or <math>a_i < \frac{P}{2}</math>. Stated in another way, it says that in every polygon, each side must be smaller than the [[semiperimeter]]. |
Revision as of 13:41, 17 August 2006
The Triangle Inequality says that in a nondegenerate triangle 
:
That is, the sum of the lengths of any two sides is larger than the length of the third side. In degenerate triangles, the strict inequality must be replaced by "greater than or equal to."
The Triangle Inequality can also be extended to other polygons. The lengths 
can only be the sides of a nondegenerate 
-gon if 
for 
. Expressing the inequality in this form leads to 
, where 
is the sum of the 
, or 
. Stated in another way, it says that in every polygon, each side must be smaller than the semiperimeter.
Example Problems
Introductory Problems
See Also
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