Polygon triangulation

Remarks

Example: let $P$ be the square with vertices $(0,0)$, $(1,0)$, $(1,1)$, $(0,1)$. The diagonal from $(0,0)$ to $(1,1)$ splits it into two triangles. Their union is $P$, their vertices are vertices of $P$, and they meet in a common side, so all three clauses hold.

Non-example: split the same square into four triangles by joining the centre $(1/2, 1/2)$ to the four corners. Clauses 1 and 3 hold, but clause 2 fails, since the centre is not a vertex of $P$.

Every simple polygon with $n$ vertices has a triangulation, and every triangulation has $n - 2$ triangles and $n - 3$ diagonals. A 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.