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Q: What is it called when a point lies in the same plane?

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That's called a "circle".

I believe this is possibly called a vector. They begin at the same point, however, they do not run on the same plane, nor do the intersect. As I said, they do begin at the same point and form a triangle.

They must be the same identical line, otherwise it's impossible.

This is the definition of a circle

A point lies on a line if the coordinates of the point satisfy the equation of the line.

a truthfull point

ray

If they are in the same plane, then they share a common plane. Did you mean to say common point. If that's the case where they are in the same plane, but do not share a common point, then they are parallel lines.

circle, but the given end point is called the center of a circle : )

Points that are within the same plane are called co-planer.

Coplanar.

A circle

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