Triangle Inequality
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.
Problems
Introductory Problems
Olympiad Problems
- Belarus 2002 Aops Topic
Given , prove:
![$\sqrt{(a+c)^2+(b+d)^2}+\frac{2|ad-bc|}{\sqrt{(a+c)^2+(b+d)^2}}\geq \sqrt{a^2+b^2}+\sqrt{c^2+d^2} \geq \sqrt{(a+c)^2+(b+d)^2}$](http://latex.artofproblemsolving.com/9/a/2/9a2b7ccc620c80325ce20ae5b267d3f8f51a483c.png)
See Also
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