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For a positive integer $n\geq 3$, the sides and diagonals of a convex $n$-gondivide the interior of the convex $n$-gon into finitely (polynomial in $n$)many regions bounded by them. In this article, we associate to every region aunique $n$-cycle in the symmetric group $S_n$ of a certain type (defined as$2$-standard consecutive cycle) by studying point arrangements in the plane.Then we find that there are more (exponential in $n$) number of such cyclesleading to the conclusion that not every region labelled by a cycle appears inevery convex $n$-gon. In fact most of them do not occur in any given singleconvex $n$-gon. Later in the main theorem of this article we characterizecombinatorially those cycles (defined as definite cycles) whose correspondingregions occur in every convex $n$-gon and those cycles (defined as indefinitecycles) whose corresponding regions do not occur in every convex $n$-gon. As aconsequence we characterize those one point extensions of a uniform rank $3$convex oriented matroid for which the one point extension is reducible by the,one point, when it lies inside the convex hull.

On the Reduciblity of a Certain Type of Rank 3 Uniform Oriented Matroid by a Point