I am taking course in Finite Element Method, and we are looking at barycentric coordinates for triangles. It appears that there is a relation between the coordinates for point in triangle of order n and triangle of order n + 1. Given by this:
$$L_1^{(n)} = \frac{S_{P23}}{S_{123}}\\ L_1^{(n+1)} = \frac{S_{P2^*3}}{S_{1^*2^*3}}\\ \frac{L_1^{(n)}}{L_1^{(n+1)}}=\frac{S_{P23}}{S_{P2^*3}}*\frac{S_{1^*2^*3}}{S_{123}}=\frac{n}{n+1}*\left(\frac{n+1}{n}\right)^2=\\ =\frac{n+1}{n} \Rightarrow L_1^{(n)}=\frac{n+1}{n}*L_1^{(n+1)} $$
Where P is a point which lies in the triangles. So my question is how do we get this equation: $$\frac{S_{P23}}{S_{P2^*3}}*\frac{S_{1^*2^*3}}{S_{123}}=\frac{n}{n+1}*\left(\frac{n+1}{n}\right)^2$$