Is correlation (in some sense) transitive?

Correlations are cosines of angles in $L^2$ hence, if $\varrho_{AB}\geqslant 0$ and $\varrho_{BC}\geqslant 0$, then $$ \varrho_{AC}\geqslant\varrho_{AB}\varrho_{BC}-\sqrt{1-\varrho_{AB}^2}\cdot\sqrt{1-\varrho_{BC}^2}. $$ Thus, if $\varrho_{AB}\geqslant c$ and $\varrho_{BC}\geqslant c$ with $c\geqslant0$, then $$ \varrho_{AC}\geqslant2c^2-1. $$ For example, if $\varrho_{AB}\geqslant90\%$ and $\varrho_{BC}\geqslant90\%$ then $\varrho_{AC}\geqslant62\%$.

I am afraid this is a non-answer: probably.

But imagine tossing a fair coin $100$ times. Let $A$ be the number of heads in the first $50$ tosses, $B$ the number of heads in the full $100$ tosses, and $C$ the number of heads in the last $50$ tosses.

Then $A$ and $B$ are (weakly) positively correlated, as are $B$ and $C$, but $A$ and $C$ have correlation $0$.

By working a little harder, we could even get negative correlation between $A$ and $C$.