The intersection of three planes can be a plane (if they are coplanar), a line, or a point.
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No, perpendicular planes intercept at only one point. Parallel planes do not intersect at all.
No, they can intersect at infinitely many points.
what of a triangle has three line segments that intersect only at their endpoints
Two planes that intersect do that at a line. neither a segment that has two endpoints or a ray that has one endpoint.
Assuming that the none of the lines are parallel, they can intersect (pairwise) at three points. Otherwise, the question is tautological.