Question 123
Let $A_1 A_2 \dots A_n$ be a regular polygon with circumcircle $C$. For each direction $d$, denote $B_k$ the second intersection with $C$ of the line of direction $d$, passing through $A_k$, $k = 1, 2, \dots, n$. Prove that the sums
$$
S = \sum_{k=1}^{n} \overrightarrow{A_k B_k}, \quad \sigma = \sum_{k=1}^{n} |\overrightarrow{A_k B_k}|^2
$$
do not depend on $d$.
Vasile Pop
$$
S = \sum_{k=1}^{n} \overrightarrow{A_k B_k}, \quad \sigma = \sum_{k=1}^{n} |\overrightarrow{A_k B_k}|^2
$$
do not depend on $d$.
Vasile Pop