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A plane midway between the two given planes and parallel to them.
Given a line, there are an infinite number of different planes that it lies in.
The given description fits that of a cylinder
Infinitely many planes contain any two given points- it takes three (non-collinear) points to determine a plane.
If 2 points determine a line, then a line contains infinitely many planes.
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There is a subtle distinction between Euclidean, Hilbert and Non-Euclidean planes. Euclidean planes are those that satisfy the 5 axioms, while Non-Euclidean planes do not satisfy the fifth postulate. This means that in Non-Euclidean planes, given a line and a point not on that line, then there are two (or more) lines that contain that point and are parallel to the original line. There are geometries where there must be exactly one line through that point and parallel to the original line and then there are also geometries where no such line contains that point and is parallel to the original line.Basically, the fifth postulate can be satisfied by multiple geometries.
There's no limit to the number of them. A more cool and sexy way to say the same thing might be: "An infinite number."
The Playfair Axiom (or "Parallel Postulate")
Euclid's parallel postulate.
Euclidean geometry is the study of points, lines, planes, and other geometric figures. The most prolonged argument over time has been that of the parallel postulate which states: there can only be one line that contains a given point and is parallel to another line.