Which shape is formed by the locus of all points equidistant from two given points?

The shape formed by the locus of all points equidistant from two given points is indeed the perpendicular bisector of the line segment that connects those two points. This is because the perpendicular bisector is defined as the line that is perpendicular to a segment and divides it into two equal halves.

When you consider a segment between two points, every point along the perpendicular bisector is at the same distance from both endpoints. This geometric property holds true because if you take any point on the perpendicular bisector, by definition, it will have equal lengths to both original points, satisfying the condition of equidistance.

In contrast, a circle is defined as the locus of points equidistant from a single point (the center), not two points. An ellipse represents a set of points where the sum of the distances to two fixed points is constant, which is a different concept altogether. A triangle is a polygon formed by connecting three points and does not represent equal distance properties associated with two specific points.

Thus, the perpendicular bisector is precisely the geometric figure that represents every point where the distance to the two given points is exactly equal.