Editing Triangle inequality (section)

===Generalization to any polygon===
The triangle inequality can be extended by [[mathematical induction]] to arbitrary polygonal paths, showing that the total length of such a path is no less than the length of the straight line between its endpoints. Consequently, the length of any polygon side is always less than the sum of the other polygon side lengths.

====Example of the generalized polygon inequality for a quadrilateral====
Consider a quadrilateral whose sides are in a [[geometric progression]] and let the sides be {{math|''a'', ''ar'', ''ar''<sup>2</sup>, ''ar''<sup>3</sup>}}. Then the generalized polygon inequality requires that

:<math>\begin{array}{rcccl} 
  0 &<& a &<& ar+ar^2+ar^3 \\
  0 &<& ar &<& a+ar^2+ar^3 \\
  0 &<& ar^2 &<& a+ar+ar^3 \\
  0 &<& ar^3 &<& a+ar+ar^2. 
\end{array}</math>

These inequalities for {{math|''a'' > 0}} reduce to the following

:<math> r^3+r^2+r-1>0 </math>
:<math> r^3-r^2-r-1<0. </math><ref>{{cite journal|title=input: ''solve 0<a<ar+ar<sup>2</sup>+ar<sup>3</sup>, 0<ar<sup>3</sup><a+ar+ar<sup>2</sup>'' |last=Wolfram{{!}}Alpha|journal=Wolfram Research|url=http://www.wolframalpha.com/input/?i=solve%20{0%3Ca%3Ca*r%2Ba*r^2%2Ba*r^3%2C%200%3Ca*r^3%3Ca%2Ba*r%2Ba*r^2}&t=ff3tb01|access-date=2012-07-29}}</ref>
The left-hand side polynomials of these two inequalities have roots that are the [[Generalizations of Fibonacci numbers#Tribonacci numbers|tribonacci constant]] and its reciprocal. Consequently, {{mvar|r}} is limited to the range {{math|1/''t'' < ''r'' < ''t''}} where {{mvar|t}} is the tribonacci constant.

====Relationship with shortest paths====
[[File:Arclength.svg|300px|thumb|The arc length of a curve is defined as the least upper bound of the lengths of polygonal approximations.]]
This generalization can be used to prove that the shortest curve between two points in Euclidean geometry is a straight line.

No polygonal path between two points is shorter than the line between them. This implies that no curve can have an [[arc length]] less than the distance between its endpoints. By definition, the arc length of a curve is the [[least upper bound]] of the lengths of all polygonal approximations of the curve. The result for polygonal paths shows that the straight line between the endpoints is the shortest of all the polygonal approximations. Because the arc length of the curve is greater than or equal to the length of every polygonal approximation, the curve itself cannot be shorter than the straight line path.<ref>{{cite book|title=Numbers and Geometry|author=John Stillwell|author-link=John Stillwell|year=1997|publisher=Springer|isbn=978-0-387-98289-2|url=https://books.google.com/books?id=4elkHwVS0eUC&pg=PA95}} p. 95.</ref>