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Good afternoon, I am trying to prove that if two planes intersect in more than one line, then they are identical:

Planes $P_1$ and $P_2$ intersect at $l_1$ and $l_2$. By the definition of a line $l_1$ contains $\alpha$ and $\beta$ and $l_2$ contains points $A$ and $B$. Therefore, $P_1$ contains $\alpha, A, B$ and $P_2$ contains $\alpha, A, B$. As any three distinct noncollinear points determine exactly one plane, $P_1 = P_2$. qed

The trouble I am having is that the definition of collinear I am given is that three or more distinct points are collinear if they lie on the same line. It does not mention anything for two points. For the definition of a plane it states that any three distinct noncollinear points determine exactly one plane. So, I am questioning to what extent my final step is valid because $A$ and $B$ do lie on the same line.

Yar
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  • "collinear" means, as you mentioned, "lying on the same line"; since if you take any two points you can draw a line they both lie on, it doesn't make sense to make the distinction between a "collinear" or "noncollinear" pair of points: in a way, they're always collinear. – sortai Oct 29 '19 at 23:02
  • While A and B do lie on the same line, since any three points taken from the set {A, B, α, β} do not, you can say they're a noncollinear set of points – sortai Oct 29 '19 at 23:03

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