Question 109
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 $P_1P_2P_3$, 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 $S'$ of the set $\{(x,y): x \ge 0, y \ge 0\}$ such that if $A \in S$ and $A' \in S'$ are the unique choices of points in (ii), then the product $BA \cdot BA'$ is a constant independent of the triangle $P_1P_2P_3$.
(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 $P_1P_2P_3$, 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 $S'$ of the set $\{(x,y): x \ge 0, y \ge 0\}$ such that if $A \in S$ and $A' \in S'$ are the unique choices of points in (ii), then the product $BA \cdot BA'$ is a constant independent of the triangle $P_1P_2P_3$.