If A and B are correlated and A and C are correlated. Why is it possible for B and C to be uncorrelated?
Imagine a random point on a plane with coordinates $(x, y)$, where $x, y \in [-1, 1]$.

A = both $x$ and $y$ are positive

B = $x$ is positive

C = $y$ is positive

It is clear A is correlated with both B and C, which are not themselves correlated (assuming uniform distribution).