One. Let z=a+bi, and from | z- 1 |a- 1+bi |= 1, that is, the square of (a- 1)+the square of b = 1 ...

1/z =1/(a+bi) = (a-bi)/(a-bi) = (a-bi)/(the square of a+the square of b). If z+ 1/z is a real number, the imaginary part of z+ 1/z, that is, b-b/(the square of a+the square of b) must be zero. After simplification, the square of A+the square of B = 1...(2)

Simultaneous (1)(2) formula, a= 1/2, b= root number 3/2 or b=- root number 3/2.