First, a GPS satellite cannot actually tell the receiver how far away it is. The receiver can detect the difference between the distance to one satellite A and another satellite B. As a result, four satellites are needed to find your position (however, if there are only three satellites then GPS can guess that you might be on the surface of the earth and calculate your location on that assumption. And if you are standing on the street, the location would be right. If you are on the tenth floor of a high building, you might be 100 metres off. If you are in a flying airplane, it's totally wrong).
Let's ignore that. If three satellites could tell you the distance from each satellite in 3D space, then there are usually two solutions to the equations. (Same in 2D space with two satellites). Typically, one solution will be close to the earth surface, and the other one won't. You pick the one close to the earth surface.
You could also do the same measurement again 100 ms later. So the two solutions move. The solution for the point where you really are moves with the speed of earth rotation. (When you sit still on your sofa, you are actually moving at about 1700 km/h or so; the number is from memory and may be wrong but it's roughly that). GPS can calculate how fast a solution moved relative to the earth surface; that speed will be low (say 190mph if you are travelling on Eurostar). The other incorrect solution will move at a massively different speed, mostly because the satellites themselves are moving at a high speed.
If you have another satellite (the third one in a 2D situation in your example), then the satellites are actually arranged in a pattern around the earth so the situation that you described will not happen. So all the circles will have one point in common. Actually, because all your data is slightly imprecise, all the circles will meet at slightly different points very close together. And you can mathematically find the point that is closest to all the circles.
And interestingly, it is illegal to build a GPS system that could be used to guide ballistic missiles. If a GPS system detects that it is at high altitude and/or moving at very high speed, as it would be typical for a ballistic missile, then it is not allowed to tell its location. You can bet that the second solution will be far from the earth surface and moving fast, so if it was the correct location it would be illegal for the GPS to report that location.