Balanced and Centre-Free Sets of Points
Jelly, 36 bytes

ạþE€Ṁ
®Œcç€®Ạ
A½×Ṡ×1,ıSµ€©œc3ñ€®Ṁ¬¢,


A full program printing the representation of a list of two ones/zeros.

This relies on theoretical infinite-precision floating point arithmetic, which is, of course, not the case. So here is a (43 byte) version which uses a 1e-10 leeway for the pair-wise distance comparisons it makes:

ạþI€×ȷ10ỊẠ€Ṁ
®Œcç€®Ạ
A½×Ṡ×1,ıSµ€©œc3ñ€®Ṁ¬¢,


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How?

(I'll include the replacement code used in place of E - equals to allow the leeway here)

ạþI€×ȷ10ỊẠ€Ṁ - Link 1: any all-equal distance complex numbers?: list of lists; list
ạþ           - outer-product with absolute difference
  I€         - incremental differences for €ach
    ×ȷ10     - multiply by 10^10
        Ị    - insignificant (abs(z)&lt;=1) (vectorises)
         Ạ€  - all? for €ach
           Ṁ - maximum

®Œcç€®Ạ - Link 2: isBalanced? no-arguments (requires register to be populated - see Main)
®       - recall from register (the points as a list of complex numbers)
 Œc     - pairs
     ®  - recall from register (the points as a list of complex numbers)
   ç€   - call the last link (1) as a dyad for €ach pair
      Ạ - all truthy?

A½×Ṡ×1,ıSµ€©œc3ñ€®Ṁ¬¢, - Main link: list of (x, y) inputs
         µ€            - for €ach (x,y) pair, p:
A                      -   absolute value of p (vectorises)
 ½                     -   square root (vectorises)
   Ṡ                   -   sign of p (vectorises)
  ×                    -   multiply (vectorises)
     1,ı               -   literal [1, 0+1i]
    ×                  -   multiply (vectorises)
        S              -   sum
           ©           - (copy the result to the register)
            œc3        - combinations of length 3
                 ®     - recall from register
               ñ€      - call the next link (1) as a dyad for €ach triple
                  Ṁ    - maximum (1 if any triple has a centre)
                   ¬   - not (0 if any triple has a centre - hence isCentreFree?)
                    ¢  - call the last link as a nilad
                     , - pair with the calculated isCentreFree?
                       - implicit print