In geometry, the plane and the point are two of the undefined terms. The other undefined term is the line. They are called as such because they are so basic that you don't really define them. They are used instead to define other terms in geometry. However, you can still describe them. A plane is a flat surface with an area of space in one dimension. A point is an indication of location. It has no thickness and no dimensions.
A plane and a point may intersect, but not always. Therefore, the correct term to be used is 'sometimes'. See the the diagram in the attached picture. There are two planes as shown. Point A intersects with Plane A, while Plane B intersects with point B. However, point A does not intersect with Plane B, and point B does not intersect with plane A. This is a perfect manifestation that a plane and a point does not always have to intersect with each other.
