= Triangulation of a convex polygon
= Triangulations of a convex polygon
{synonym}
A <triangulation of a convex polygon> divides a <convex polygon> into <triangles> using diagonals between its existing <vertices of a polytope>, with no diagonal crossings in the interior. For $m\geq3$ vertices it uses $m-3$ diagonals and produces $m-2$ <triangles>. Fixing one boundary edge, its adjacent <triangle> separates two smaller polygon regions; summing over the possible third <vertices of a polytope> gives the <Catalan recurrence>. Thus the number of <triangulations of a convex polygon> is $C_{m-2}$, with $C_n$ the <Catalan number>. A quadrilateral has two choices and a pentagon has five.
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