MathDB

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

\text{(i)}

there is a point

T

S

such that for every point

Q

S

, the segment

TQ

lies entirely in

S

; and

\text{(ii)}

for any triangle

P_1P_2P_3

, there exists a unique point

A

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 \geq 0, y \geq 0\}

such that if

A \in S

and

A' \in S'

are the unique choices of points in

\text{(ii)}

, then the product

BA \cdot BA'

is a constant independent of the triangle

P_1P_2P_3

G3