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So is "geometry" maths-speak for "shape"?

And does "flat" quite literally mean flat, like a 2D-sheet?

And if that's true, how can the rule of homogeneity apply?

Within a two-dimensional plane, for any given line ℓ and a point A, which is not on ℓ, there is exactly one line through A that does not intersect ℓ.