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One definition of a plane says a non-curved surface that includes three non-co-linear points. Ok fine. But how can we define a non-curved surface using simple constructions without relying on the assumed idea of non-curved.

It's one thing to say a non-curved surface that includes 3 non-co-linear points but how can the 'flatness' of the surface we refer to as non-curved be defined using only geometric constructions? What makes flatness using only geometric terms?

Here's my try...a surface that includes 3 non-co-linear points, A, B, and C, and also includes all of the points contained by the three lines a, b, and c, passing through each pair of points A, B, and C, as well as including all of the points contained by all of the lines passing through any point on one of those three lines a, b, or c, and any other point on either of the other two original lines.

This seems to lay out three points...create 3 lines and an infinity of other lines all in a 2 dimensional plane. There are only one problem, how can we define the 'straightness' of a line (which is necessary to assume in my definition) by using only geometric constructions and that doesn't require the use of the term 'one dimensional) or assume an axiomatic concept of 'straightness'? This even requires the definition of linear. How do you define linear using only geometric construction and how can we define a plane?

In defining a sphere, for example, we can say that it is an object made up of only those points equidistant from a reference point. This is a self-contained definition that relies only on geometric construction. How do we define a line or plane in the same way?

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