Let P(n) denote the number of possibilities. For a triangle (n=3) we obviously have P(3)=1.

Suppose that we have determined P(k) for values k<n, and proceed by induction on n.

Without loss of generality, we can suppose that A₁ A₂ is always a side of one of the resulting triangles in the partition of the n-gon. The third vertex of the triangle must be one point among A₃ ,…, A_n.

The number of partition possibilities for the (n-1)-gon A₁, A₃,… ,A_n is P(n-1). If the third vertex is A₄, then number of possibilities is that of the (n-2)-gon A₁, A₄,… ,A_n added to those of A₂, A₃,A₄, i.e., P(n-2)P(3). If the third vertex is A₅, then the number is given by P(n-3)P(4), corresponding to the possibilities for the (n-3)-gon A₁, A₅,… ,A_n added to those of A₂, A₃, A₄,A₅ . Hence for A_k as third vertex, the number of possibilities is given by P(n+2-k)P(k-1), from which:

P(n) =  P(n-1) + P(n-2)P(3) + P(n-3)P(4)+ P(n-4)P(5) + … +  P(n-3)P(4)+   P(n-2)P(3) +   P(n-1)

From this relation the exact number in dependence of n can be derived as:

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