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 is only one point that will be equidistant from three other points. It is the point where the three (or any two of the three, of course) perpendicular bisectors of the segments between the pairs of points intersect.

For example, consider points P (0,0), Q (6,0), and R (10,4). The midpoint of PQ is (3,0), and the perpendicular bisector is x = 3. The midpoint of QR is (8,2) and the perpendicular bisector is y = -x + 10. The bisectors intersect at (3,7) and the distance between that point and P, Q, or R is the square root of 58.