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Any three points make up a plane. Think about a plane as a sheet of paper; this is how I visualize it. If three atoms are on the same plane, that means that the three atoms can be attached to each other through a "sheet of paper".
Yes, this is a good example^. I also like to visualize the atomic models that Dr. Lavelle had in class when discussing shapes. Being on the same plane means that a certain number of points are flat. So, trigonal planar, for example, all 3 atoms are in the same plane. We could see that in the model because it was a "flat" triangle. However, when we have something like octahedral, there are 4 atoms on the same plane that form a "flat square" and two atoms that stick out from the square vertically. Those two are now in a different plane because they don't lay flay with the others.
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