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Given $a, b​​, c, d>0$. ¿What is the necessary and sufficient condition so that it can form a convex quadrilateral with sides $a, b​​, c, d$?

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up vote 3 down vote accepted

Note that convexity is irrelevant, since a non-convex quadrilateral can be transformed into a convex quadrilateral by reflecting two sides from the interior of the convex hull to the outside.

The necessary and sufficient condition is analogous to the one for the sides of a triangle, the triangle inequality: $a$, $b$, $c$ and $d$ can form a quadrilateral if and only if the sum of any three of them is greater than the fourth.

To see this, imagine one side fixed and two adjacent sides rotatable about the vertices they share with the fixed side. If both adjacent sides are extended away from the fixed side, the distance between their free ends is the sum of the three sides. Now rotate the two sides inward. If the sum of their lengths is greater than the length of the fixed side, their free ends can be brought to coincide. If not, their distance can be reduced to the difference between the fixed side's length and the sum of the lengths of the adjacent sides. Thus, the distance of the free ends will at some point of the movement be equal to the length of the fourth side if and only if the above condition is fulfilled.

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