NL-NWO 1989 Problem 4

Given is a regular $n$ -sided pyramid with top $T$ and base $A_1A_2A_3... A_n$ . The line perpendicular to the ground plane through a point $B$ of the ground plane within $A_1A_2A_3... A_n$ intersects the plane $TA_1A_2$ at $C_1$ , the plane $TA_2A_3$ at $C_2$ , and so on, and finally the plane $TA_nA_1$ at $C_n$ . Prove that $BC_1 + BC_2 + ... + BC_n$ is independent of choice of $B$ 's.