If $\operatorname{Corr}(X,Y)=1$, then $ \operatorname{Corr}(X,Z)=\operatorname{Corr}(Y,Z)$

Question

$\DeclareMathOperator{\Corr}{Corr}$Given that I have three r.v. $X,Y,Z$ and $\Corr(X,Y)=1$, can I then conclude that $\Corr(X,Z)=\Corr(Y,Z)$?

I've tested on some data and found that it was true in my tests, but I've had no luck in regards to the math.

Answer 1

Since $\operatorname{Corr}(X,\,Y)=1$, constants $a,\,b$ exist with $Y=aX+b,\,a\gt0$. Then$$\operatorname{Corr}(X,\,Z)=\operatorname{Corr}(aX,\,Z)=\operatorname{Corr}(aX+b,\,Z)=\operatorname{Corr}(Y,\,Z).$$