A GRE Geometry Question

Question

P, Q, and R are three points in a plane, and R does not lie on line PQ.
Which of the following is true about the set of all points in the plane that
are the same distance from all three points?
A It contains no points.
B It contains one point.
C It contains two points.
D It is a line.
E It is a circle.

Please Help The answer is B

Only thing I can imagine is that point R must be anywhere on the perpendicular bisector of line PQ except the point where it intersects with the line PQ.

Answer 1

There are many ways to do this, but I have found that when dealing with GRE questions, you want to solve them quickly and in a minimal fashion, since it's multiple choice. Consider then the special case where R forms an equilateral triangle. The center of mass is the one point lying equidistant from P Q and R. Clearly though there is no other such solution. The answer must be B then by default.

Answer 2

Notice that we can construct an unique circle through ANY 3 noncollinear points.

In light of above information, do you see what point could be located at same distance from the given three points ?

Answer 3

Is the center of this circle, the intersection of mediators of sides of the triangle . $$OP=OQ=OR$$.