NMinimize the sum of the squares of the left hand side of your twelve equations quickly finds a reasonably small minimum. Perhaps casting your problem in terms of minimization will get you what you want.

You should replace the exponents 2.0 with 2, so that it is an algebraic system instead of a transcendent system. However, you cannot solve this system exactly, as it is overdetermined. For example, eq1, eq3, eq5, eq7 are four circles that do not intersect all together:

ContourPlot[Evaluate[{eq1, eq3, eq5, eq7}],
 {xA, -8, 5}, {yA, -8, 5.5}]