Suppose We are given the length of all six sides of a Convex Hexagon. How can we tell whether it's valid or Not ? that means can we tell whether it's area is positive or not ?
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$\begingroup$ I'm not sure why this was edited just now - it is clear from JoR's answer that if the longest edge is equal to the sum of the others, the area is zero. On the other hand, if the area is zero, it does not imply that this condition holds, for example all edges could be the same length. $\endgroup$– user25199Commented Apr 30, 2014 at 6:38
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$\begingroup$ I interpreted the question differently in my answer, but I see now that other interpretations are possible. $\endgroup$– Joseph O'RourkeCommented Apr 30, 2014 at 15:39
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A chain of edges can close iff the longest edge is not longer than the sum of the lengths of all the other edges. This is Theorem 8.6.3 (p.326) in Computational Geometry in C and Theorem 5.1.2 (p.61) in Geometric Folding Algorithms. You can easily see the necessity of this condition: If one edge $e$ is too long, the others all together cannot reach from end-to-end of $e$.
This result has nothing to do with $6$; it holds for $n \ge 3$ edges.
answered May 23, 2013 at 18:40