Is it true that $\text{min}\{x/2, y/2\} = \frac{1}{2}\text{min}\{x, y\}$?

Is it true that $ \text{min}\{x/2, y/2\} = \frac{1}{2}\text{min}\{x, y\}$ ? Or more generally, $ \text{min}\{cx, cy\} = c \cdot \text{min}\{x, y\}$ ?

I think that the answer is YES. But, I got this guess by just plugging in a few numbers. Maybe I am missing some sort of clever counterexample. Can someone please help me?