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Here is one option: 2 points uniquely define a line so a line can be named after any two points that belong to it. Similarly, three points that are not collinear (all in the same line) uniquely define a plane so a plane can be defined by naming any three non-collinear points in it.

There are different - though related - forms in coordinate geometry or in vector algebra.

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Q: How can you name a line and a plane?
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