Prove that if a fraction is broken up into two the resulting two fractions cannot both have a larger value

Question

I am working on research involving probability tables. I simplified the problem to the following. Say we have the following:

$C_1, C_2, C_3, C_4$

$\forall i, C_i > 0$

$x = \frac{C_1 + C_2}{C_1 + C_2 + C_3 + C_4}$

$y = \frac{C_1}{C_1 + C_3}$

$z = \frac{C_2}{C_2 + C_4}$

I am almost positive (by just filling in numbers) that both $y$ and $z$ cannot $> x$ or $< x$. Meaning that unless $y = z$, then either $y < x$ and $z > x$ or $y > x$ and $z < x$. Does anyone have suggestions for an approach to prove this?

==EDIT== Basically, I am trying to prove that either of these two situations are impossible:

$z \geq y > x$

$x > y \geq z$

Answer 1

The number $x$ defined in this way is called the mediant of the two numbers $y$ and $z$ defined in this way. It is known that the mediant of two distinct numbers is strictly between those numbers. See this question and its answers.

(This answer replaces my previous answer completely, as that answer was based on a slight misformatting of the question, which has since been corrected.)