Question 99
Let $B = (-1, 0)$ and $C = (1, 0)$ be fixed points in the plane. A non-empty, bounded subset $S$ of the plane is called nice if the following conditions hold:
(i) There is a point $T$ in $S$ such that for every other point $Q$ in $S$ the segment $TQ$ lies entirely in $S$.
(ii) For every triangle $P_{1} P_{2} P_{3}$ there exist a unique point $A$ in $S$ and a permutation $\sigma$ of $\{1,2,3\}$ such that the triangles $ABC$ and $P_{\sigma(1)} P_{\sigma(2)} P_{\sigma(3)}$ are similar.
Show that there are two different nice subsets $S$ and $S'$ of the set $\{(x, y): x \geq 0, y \geq 0\}$ with the following property: the product $BA \cdot BA'$ is independent of the choice of the triangle $P_{1} P_{2} P_{3}$, where $A \in S$ and $A' \in S'$ are the unique points from (ii) for an arbitrary triangle $P_{1} P_{2} P_{3}$.
(i) There is a point $T$ in $S$ such that for every other point $Q$ in $S$ the segment $TQ$ lies entirely in $S$.
(ii) For every triangle $P_{1} P_{2} P_{3}$ there exist a unique point $A$ in $S$ and a permutation $\sigma$ of $\{1,2,3\}$ such that the triangles $ABC$ and $P_{\sigma(1)} P_{\sigma(2)} P_{\sigma(3)}$ are similar.
Show that there are two different nice subsets $S$ and $S'$ of the set $\{(x, y): x \geq 0, y \geq 0\}$ with the following property: the product $BA \cdot BA'$ is independent of the choice of the triangle $P_{1} P_{2} P_{3}$, where $A \in S$ and $A' \in S'$ are the unique points from (ii) for an arbitrary triangle $P_{1} P_{2} P_{3}$.