What are the ways 3 planes in $ℝ^3$ intersect?

Question

Will they be able to form points, lines, or planes?

In my opinion, the planes can only form points and lines, could someone give any counterexamples?

Answer 1

Two distinct planes intersect in a line or not at all (prove this!). The line can intersect a third plane in either a line or a single point (prove this!). It's easy to construct examples where each of these cases can happen.

Now if two or more of the planes coincide, there's one more case.

Answer 2

1) The three planes can be parallel. And not intersect at all.

If two planes intersect the intersection will be a line.

2) Two planes can be parallel and the third plane intersects each. The third intersects each at a line. These to lines are parallel and co-planer.

3) All planes intersect at a line and the third intersects the two on the same line (like pages in an open book intersecting at the spine).

4) The two planes intersect and a line. The third intersect each at a parallel angle to insect at second and third parallel line. The planes will from a triangular cylinder, with each pair of planes intersecting at a line. These three lines are mutually parallel and non planar.

5) General case. Two planes intersect. A third intersects obliquely and the three intersect at a point. Each pair of planes intersect at a line. The three lines are neither coplanar nor parallel and the three lines intersect at the point where the three planes do.