Let and
be fixed points on the coordinate plane. A nonempty, bounded subset
of the plane is said to be nice if
there is a point
in
such that for every point
in
, the segment
lies entirely in
; and
for any triangle
, there exists a unique point
in
and a permutation
of the indices
for which triangles
and
are similar.
Prove that there exist two distinct nice subsets and
of the set
such that if
and
are the unique choices of points in
, then the product
is a constant independent of the triangle
.