Question 148
In the coordinate plane there are two fixed points $B(-1, 0)$, $C(1, 0)$. A bounded non-empty subset $S$ of the plane is called a "good set" if it satisfies the following two conditions:
(i) there is a point $T$ in $S$ such that for every point $Q$ in $S$, the segment $TQ$ lies entirely in $S$;
(ii) for every triangle $P_1P_2P_3$, one can find a unique point $A$ in $S$ and a permutation $\sigma$ of the set $\{1, 2, 3\}$ such that triangle $ABC$ is similar to $P_{\sigma(1)}P_{\sigma(2)}P_{\sigma(3)}$.
Prove that within the set $\{(x, y): x \ge 0, y \ge 0\}$ there exist two different good sets $S, S'$ such that: if $A \in S$ and $A' \in S'$ are the unique choices satisfying condition (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 $S$;
(ii) for every triangle $P_1P_2P_3$, one can find a unique point $A$ in $S$ and a permutation $\sigma$ of the set $\{1, 2, 3\}$ such that triangle $ABC$ is similar to $P_{\sigma(1)}P_{\sigma(2)}P_{\sigma(3)}$.
Prove that within the set $\{(x, y): x \ge 0, y \ge 0\}$ there exist two different good sets $S, S'$ such that: if $A \in S$ and $A' \in S'$ are the unique choices satisfying condition (ii), then the product $BA \cdot BA'$ is a constant independent of the triangle $P_1P_2P_3$.