Q: What are points called if there is a plane that contains all of them?

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collinear

The set of all the points is the plane itself.

They are called equidistant points and form points on a sphere for a solid or a circle on a plane figure.

It's a plane. I think

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.

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collinear

A group of points that extend forever in all directions is called a plane. We often draw a plane with edges, but it really has no edges.

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

Only if the 3 points are all in the same line. Then there are an infinite number of planes.If the 3 points are not all in the same line, then there is only one unique plane that contains them.That's what "define" means.

The set of all the points is the plane itself.

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

They are called equidistant points and form points on a sphere for a solid or a circle on a plane figure.

It's a plane. I think

A line segment (sometimes just segment) is a pair of endpoints and all the points on a line between them.

A line segment (sometimes just segment) is a pair of endpoints and all the points on a line between them.

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.

I believe that's a circle...