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Q: Where do all the points on a plane equidistant from the endpoints of ab lie?

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All of the points on a perpendicular bisector are equidistant from the endpoints of the segment.

All points on the circumference of a circle drawn on a plane are equidistant from the single point on the plane which is the center of the circle.

J

A Circle.

line

Related questions

The set of all points in the plane equidistant from one point in the plane is named a parabola.

All of the points on a perpendicular bisector are equidistant from the endpoints of the segment.

All points on the circumference of a circle drawn on a plane are equidistant from the single point on the plane which is the center of the circle.

J

That's a circle around the center, in the plane.

A Circle.

line

Math

That set of points forms what is known as a "circle".

Bisector of an angle, is defined as the set of all points in a plane that are equidistant from the two sides of a given angle.

A plane is the set of all points in 3-D space equidistant from two points, A and B. If it will help to see it, the set of all points in a plane that are equidistant from points A and B in the plane will be a line. Extend that thinking off the plane and you'll have another plane perpendicular to the original plane, the one with A and B in it. And the question specified that A and B were in 3-D space. Another way to look at is to look at a line segment between A and B. Find the midpoint of that line segment, and then draw a plane perpendicular to the line segment, specifying that that plane also includes the midpoint of the line segment AB. Same thing. The set of all points that make up that plane will be equidistant from A and B. At the risk of running it into the ground, given a line segment AB, if the line segment is bisected by a plane perpendicular to the line segment, it (the plane) will contain the set of all points equidistant from A and B.

In spherical geometry we look at the globe as the sphere S^2. Any plane intersecting the sphere will create a great circle. Now if you take any point on the globe and reflect it across that plane, you have another point that is equidistant from the plane. The sets of all these points will be equidistant from the great circle.