Can corr(A,C) be 0.1 given the other two

Suppose $B$ has correlation $0.9$ with $A$ and correlation $0.8$ with $C$. Could the correlation between $A$ and $C$ be as low as $0.1$?

No. The $3\times3$ correlation matrix must be positive semi-definite, which forces $\rho_{AC}\in\left[0.72-\sqrt{(1-0.81)(1-0.64)},\,0.72+\sqrt{(1-0.81)(1-0.64)}\right]\approx[0.46,\,0.98]$. With $\rho_{AC}=0.1$ the determinant is negative.