Determine whether there exist two infinite point sequences $ A_1,A_2,\ldots$ and $ B_1,B_2,\ldots$ in the plane, such that for all $i,j,k$ with $ 1\le i < j < k$, (i) $ B_k$ is on the line that passes through $ A_i$ and $ A_j$ if and only if $ k=i+j$. (ii) $ A_k$ is on the line that passes through $ B_i$ and $ B_j$ if and only if $ k=i+j$. (Proposed by Gerhard Woeginger, Austria)