I don't understand why Reduce works for this problem but Solve doesn't.
In[15]:= Solve[{Abs[z] == 1, Abs[z - 1/2] == 1}, z, Complexes] Out[15]= {} In[17]:= Reduce[{Abs[z] == 1, Abs[z - 1/2] == 1}, z] Out[17]= z == 1/4 - (I Sqrt[15])/4 || z == 1/4 + (I Sqrt[15])/4
Thanks. The documentation says
Solve gives generic solutions only. Solutions that are valid only when continuous parameters satisfy equations are removed. Additional solutions can be obtained by using nondefault settings for MaxExtraConditions.
which explains it.
Solve gives the solution if you add MaxExtraConditions -> All
Solve[{Abs[z] == 1, Abs[z - 1/2] == 1}, z, Complexes, MaxExtraConditions -> All]
Solve does not specify conditions in the solution, so quite puzzling for me
