Range of corr(y,z) when both correlate 0.8 with x

Three random variables $x,y,z$ are such that $x$ has correlation $0.8$ with $y$ and also correlation $0.8$ with $z$. Find the maximum and the minimum possible value of the correlation between $y$ and $z$.

Between $0.28$ and $1$. Think of standardised variables as unit vectors with correlation as the cosine of the angle. Each of $y,z$ is at angle $\theta=\arccos0.8$ from $x$, so their angle lies in $[0,2\theta]$ and the correlation in $[\cos2\theta,1]=[2(0.8)^2-1,1]$. The PSD condition on the correlation matrix gives the same: $0.64\pm0.36$.