IMO Team Selection Contest II

Let $B=(-1,0)$ and $C=(1,0)$ be fixed points on the coordinate plane. A nonempty, bounded subset $S$ of the plane is said to be nice if (i) there is a point $T\in S$ such that for every point $Q\in S$, the segment $TQ$ lies entirely in $S$; and (ii) for any triangle $P_1{P}_2{P}_3$, there exists a unique point $A\in S$ and a permutation $\sigma$ of the indices $\{1,2,3\}$ for which triangles $ABC$ and $P_{\sigma (1)}{P}_{\sigma (2)}{P}_{\sigma (3)}$ are similar. Prove that there exist two distinct nice subsets $S$ and $S^{\prime }$ of the set $\{(x,y):x\ge 0,y\ge 0\}$ such that if $A\in S$ and $A^{\prime }\in S^{\prime }$ are the unique choices of points in (ii), then the product $BA\cdot B{A}^{\prime }$ is a constant independent of the triangle $P_1{P}_2{P}_3$.