Given $ \mid x - y \mid = 1$ and equality predicates in $ \mathbb{C} $.How to get $ \mid x - y \mid = \frac{1}{2} $?

Question

You are given $ \mid x - y \mid = 1$ and equality predicates in $ \mathbb{C} $ . Your task is to construct the predicate $ \mid x - y \mid = \frac{1}{2} $.

Can you help me with this task?

For exmaple, I can do it if there was $ 2 $ instead of $ \frac {1}{2} $

$ \exists z (\mid x - z \mid = 1) \land(\mid y - z \mid = 1) \land (\forall a ((\mid x - a \mid = 1) \land (\mid y - a \mid = 1)) \to (z = a)$)

Similarly we can do that for any $ k \in \mathbb{N} $. I can get $ k\times\sqrt{3}$ too. I just cannot get any less distances than $ 1 $.

Answer 1

I believe this can be obtained from the geometric construction of bisecting a line segment using compasses only (no straightedge). You can find such a construction here.