# Proof 5.12

Using coordinates, write a detailed step-by-step proof that the set of points equidistant from the two fixed points A and B is the perpendicular bisector of the segment AB.
Let  be a line segment with end points  and . Denote  as  and  as  1. Using the midpoint formula, we can determine that the midpoint  is located at . 2. Consider a fourth point  equidistant from  and . Denote  as . 3. We can determine the distance from  to  using the distance formula.  4. Similarly, we can determine the distance from  to . 5. Since we know that  is equidistant from  and , we can set the formulas from 3 and 4 equal to one another and solve for . . 6. Since the x-coordinate of  is zero, we know that it lies on y-axis. Recall,  also lies on the y-axis which is perpendicular by definition to the x-axis where  and  are located. 7. Since  is the midpoint of  and lies on a line perpendicular to , the line  is the perpendicular bisector of . Therefore, the set of points equidistant from two fixed points is the perpendicular bisector of the segment formed by the fixed points.