Let $R^3$ denote the three-dimensional space. Take two points $P=(1,2,3)$ and $Q=(4,2,7)$. Let dist $(X, Y)$ denote the distance between two points $X$ and $Y$ in $R^3$. Let
$S=\left\{X \in R^3:(\operatorname{dist}(X, P))^2-(\operatorname{dist}(X, Q))^2=50\right\}$ and
$T=\left\{Y \in R^3:(\operatorname{dist}(Y, Q))^2-(\operatorname{dist}(Y, P))^2=50\right\}$
Then which of the following statements is (are) TRUE?
Select ALL correct options:
A
There is a triangle whose area is 1 and all of whose vertices are from S.
B
There are two distinct points L and M in T such that each point on the line segment LM is also in T.
C
There are infinitely many rectangles of perimeter 48 , two of whose vertices are from S and the other two vertices are from T.
D
There is a square of perimeter 48 , two of whose vertices are from S and the other two vertices are from T.
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