Let $f(K)$ be the value assigned to point $K$. Let $A$ be any point, and $AB_{21}B_{31}...B_{n1}$ be similar to $P$. Now draw $B_{21}B_{22}...B_{2n}$ similar to $P$. Now consider spiral symmetry about $A$ that maps $B_{21}B_{22}...B_{2n}$ to $B_{i1}B_{i2}...B_{in}$. Then $B_{i1}B_{i2}...B_{in}$ and $AB_{2j}B_{3j}...B_{nj}$ are similar to $P$. Summing all of $B_{i1}B_{i2}...B_{in}$ up, we have $\sum_{i=2,j=1}^n f(B_{ij})=0$. Summing all of $AB_{2j}...B_{nj}$ up, we have $\sum_{i=2,j=1}^n f(B_{ij})+nf(A)=0$. Taking the difference we have $f(A)=0$ and we are done.