Let $X, Y, Z$ be 3 scalar random variables.
Given $\rho_{X,Z}$ and $\rho_{Y,Z}$, what can we say about $\rho_{Z, X+Y}$?
What I've tried:
Recall that correlation is cosine of the angles between the random variables. In the trivial case, $X$ and $Y$ could be pointing in the exact same direction, in which case $\rho_{X,Z}$ = $\rho_{Y,Z}$ = $\rho_{Z, X+Y}$. I am guessing this happens when $X = c \cdot Y$, i.e. $X$ is $Y$ scaled by a constant or vice versa.
However, when $\rho_{X,Z} \neq \rho_{Y,Z}$ (i.e. in the non-trivial case when $X$ and $Y$ does not have the same direction), I am struggling to say anything about $\rho_{Z, X+Y}$.
Is it possible to conclude anything useful about $\rho_{Z, X+Y}$?
Suppose $X$, $Y$ and $Z$ are standardized r.v.s with variance 1. So that is true that $\text{Cov}(X,Z)=\rho_{x,z}$ and $\text{Cov}(Y,Z)=\rho_{y,z}$ and $$\rho_{z,x+y}=\dfrac{\text{Cov}(Z,X+Y)}{\sigma_{x+y}\sigma_z}=\dfrac{\text{Cov}(Z,X)+\text{Cov}(Z,Y)}{\sigma_{x+y}\sigma_z}=\dfrac{\rho_{z,x}+\rho_{z,y}}{\sigma_{x+y}\sigma_z}.$$ As $\sigma^2_{x+y}=\sigma^2_x+\sigma^2_y+2\sigma_x\sigma_y \rho_{x,y}$, and $\sigma_x=\sigma_y=\sigma_z=1$, it follows: $$\rho_{z,x+y}=\dfrac{\rho_{z,x}+\rho_{z,y}}{\sqrt{2(1+\rho_{x,y})}}\ \ \ \text{or, if}\ \ \rho_{x,y}=0,\ \ \ \ \rho_{z,x+y}=\dfrac{\rho_{z,x}+\rho_{z,y}}{\sqrt{2}}$$