Editing Triangle inequality (section)

====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.