No, it is not possible for two distinct planes to intersect at a single point.
Understanding Plane Intersections
In geometry, a plane is a flat, two-dimensional surface that extends infinitely in all directions. When two planes cross paths, the way they interact is determined by fundamental geometric principles.
According to geometric rules, including the information provided: Only lines intersect at a point. Planes intersect along a line.
This means that if two planes are not parallel and are not the same plane, their intersection will always be a straight line. Imagine two pieces of paper crossing each other; the place where they meet forms a line, not just a single dot.
Why Not a Single Point?
Consider the properties of a plane:
- Infinite Extent: A plane has no boundaries; it goes on forever.
- Flatness: It has no curves or thickness.
- Contains Lines: If two points lie in a plane, the entire line connecting those points also lies within that plane.
If two planes were to intersect at only one point, say Point P, consider any other point Q in one of the planes, different from P. If Plane 1 and Plane 2 only shared Point P, then the line segment PQ would lie entirely within Plane 1, but only Point P would potentially be in Plane 2 (it wouldn't be, as the intersection is only P). However, the nature of planes dictates that if they intersect at more than one point (and the reference tells us they don't just intersect at one point, but along a line), their intersection must be a line.
The reference confirms this:
- Lines intersect at a point. (e.g., two non-parallel lines on a flat surface)
- Planes intersect along a line. (e.g., the spine of a book where two pages meet, extended infinitely)
Comparing Intersections
| Geometric Object | Intersection Type (Typically) | Example |
|---|---|---|
| Two Lines | A Point | Corner of a square |
| Two Planes | A Line | Edge where two walls meet |
| A Line and a Plane | A Point or the Line itself | Pencil piercing a sheet of paper (point) or a line drawn on the paper (line) |
Therefore, based on the established rules of geometry and the provided information, the intersection of two distinct planes is always a line, provided they intersect at all (they could also be parallel and not intersect).
