Does higher variance imply a higher covariance?

Question

Suppose I have three random variables A,B,C. if var(B) > var(C) does that mean cov(A,B) > cov(A,C)? Assuming neither is uncorrelated meaning cov(A,B) and cov(A,C) don't equal 0.

Answer 1

Clearly not. A counterexample id given by $(A,B,C)$ multinomal distributed with covariance matrix $$ \begin{pmatrix} 1 & 0.14 & 0.9 \\ 0.14 & 2 & 0 \\ 0.9 & 0 & 1 \end{pmatrix} $$ (you can check that is a valid covariancematrix by calculating the eigenvalues, which are $ 2.06, 1.84, 0.09$, all positive)