Let $A, B, C$ be random variables with covariance $cov(A B) = 0$ and correlation $corr(B, C) = 1$.
Is it true that $cov(A C) = 0$?
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Yes, it has to be $0$. Assume $\text{corr}(A,C)=p$. Then, the correlation matrix of the three random variables is $$C=\begin{bmatrix}1&0&p\\0&1&1\\p&1&1\end{bmatrix}$$
And, this has to be positive semi-definite (PSD). A first condition of positive semi-definiteness is having non-negative determinant. Here, $\det(C)=-p^2$ which means $p$ can only be $0$. That makes also covariance $0$.
Note that, it'd have been tempting to say that $\text{corr}(B,C)=1$ implies a linear relationship between the two RVs and this makes covariance of $A$ and $C$ zero, but, this may not be the case when the probability space is infinite and the linear relationship does not hold for some $B$ and $C$ values with zero probability measure.
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3$\begingroup$ When a relationship holds except on a set of zero probability, we still say it holds (and will qualify that statement with "almost surely" in order to be clear). Thus, the argument is valid: because the correlation of $B$ and $C$ is $\pm 1,$ $B$ and $C$ are linearly related a.s.; say, $C=\alpha B + \beta.$ Then $\operatorname{Cov}(A,C)=\alpha\operatorname{Cov}(A,B)$ and the result follows as a special case. $\endgroup$– whuber ♦Commented Feb 13, 2022 at 16:16