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.
- Belarus 2002 Aops Topic
Given , prove:
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