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This question could be seen as a coordinate-free variant of Sylvester's Four Point Problem (cf e.g. http://mathworld.wolfram.com/SylvestersFour-PointProblem.html):

Suppose one are given an inifinitely thin stick of unit length, which fraction of the planar quadrilaterals you can generate by cutting the stick into six pieces (for the sides and diagonals) has a convex hull with four corners, when embedded into the euclidean plane?

In that setting, it is the number of non-similar quadrilaterals that counts, which means, that the same partitioning of the stick may yield several non-similar quadrilaterals, which are counted separately.

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    $\begingroup$ An earlier, unsolved MO question asks if the six pieces can form a tetrahedron. And a still earlier, solved question asks if breaking into three pieces forms a triangle. $\endgroup$
    – Joseph O'Rourke
    Commented Aug 16, 2014 at 13:21
  • $\begingroup$ A variant on your question would break into four pieces to form a convex quadrilateral, ignoring the diagonals. $\endgroup$
    – Joseph O'Rourke
    Commented Aug 16, 2014 at 13:22
  • $\begingroup$ @Joseph O'Rourke four segments do not define a discrete set of quadrilaterals and, deltoids would also be constructible from such a set; I asked for six pieces in order to make the resulting quadrilaterals rigid. $\endgroup$
    – Manfred Weis
    Commented Aug 16, 2014 at 14:07

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