> [!NOTE] Statement 1 \[MA241\] > The [[Catalan Numbers]] $C_{n}$ count the number of triangulations of the $(n+2)$-gon. ###### Proof: Define $C_{n}$ as the number of triangulations of the regular $(n+2)$-gon. WTS $C_{n}=\sum_{i=0}^{n-1}C_{i}C_{n-i}$. Label an edge of the $(n+2)$-gon. Label the other vertices $0,1,2,\dots,n-1$. If the triangle containing the labelled edge is triangle formed by connecting its endpoints with vertex $i$, this partitions the original polygon into an $i$-gon and and $(n-i)$-gon. $\square$.