The locus of points which are equidistant from two fixed points is

Take two fixed points A and B. A point P is the same distance from both, so PA = PB.

The midpoint of AB satisfies this. So does every point on the line through the midpoint at right angles to AB.

That line is the perpendicular bisector of AB. It is a straight line, but “a line” is too general: only this one line works.