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  "body": "\n## Definition\n\nLet $P \\subset \\R^2$ be a simple polygon, that is, a closed region bounded by a closed polygonal chain with $n \\ge 3$ vertices that does not cross itself.\nA *triangulation* of $P$ is a finite set $T$ of [[triangles|0CTNFGYMM2]] such that:\n\n1. the union of the triangles of $T$ is $P$;\n2. the vertices of each triangle of $T$ are vertices of $P$;\n3. any two distinct triangles of $T$ meet in the empty set, in one common vertex, or in one common side.\n\nEquivalently, a triangulation is given by a maximal set of diagonals of $P$ that pairwise meet only at endpoints.\nA diagonal is a [[line segment|VQ4Q2V45RP]] joining two vertices of $P$ that lies in $P$ and meets the boundary of $P$ only at its endpoints.\n\n## Remarks\n\nExample: let $P$ be the square with vertices $(0,0)$, $(1,0)$, $(1,1)$, $(0,1)$.\nThe diagonal from $(0,0)$ to $(1,1)$ splits it into two triangles.\nTheir union is $P$, their vertices are vertices of $P$, and they meet in a common side, so all three clauses hold.\n\nNon-example: split the same square into four triangles by joining the centre $(1/2, 1/2)$ to the four corners.\nClauses 1 and 3 hold, but clause 2 fails, since the centre is not a vertex of $P$.\n\nEvery simple polygon with $n$ vertices has a triangulation, and every triangulation has $n - 2$ triangles and $n - 3$ diagonals.\nA convex polygon with $n$ vertices has exactly $C_{n-2}$ triangulations, where $C_m = \\frac{1}{m+1}\\binom{2m}{m}$ is the $m$-th Catalan number.\n",
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