Why can a line never intersect a plane in exactly two points?
I know this seems like a really simple question, but I'm having a hard time figuring out how to answer it. I also tried googling the question but I couldn't find an answer for exactly what I'm looking for.
If you pick two points on a plane and connect them with a straight line then every point on the line will be on the plane.
Given two points there is only one line passing those points.
Thus if two points of a line intersect a plane then all points of the line are on the plane.
I think you're having trouble with the question because there isn't a satisfying answer. The statement "a line can never intersect a plane at exactly two points" is either an axiom in some formalization of Euclidean geometry or follows so directly from one or two other axioms in the system that the answer seems empty of meaning, a restatement of definitions (as in some of the good answers here).
An axiom is a statement that's taken as a given, and that's where mathematics starts. The question of why any particular axiom exists or is justified is to some extent a philosophy of science question. In the case of Euclidean geometry I think the answer is that the rules seem to (mostly) mirror our experience of the physical world we inhabit and the mathematical results of the system lead to useful practical results (helpful in building a house, for instance).
But there are other systems of geometry with different/fewer axioms which seem intuitively absurd yet produce useful results as well.
EDIT: see Paul Sinclair's answer
As jberryman has expressed, the real issue here is what is meant by a "line" and a "plane". Traditionally, these are taken as undefined primitive concepts, and the ideas that any two points will determine a unique line, and that if two points are in a plane, then the line through those two points will lie entirely in that plane are taken as axioms.
The problem with definitions is that they can only introduce new ideas in terms of older ideas. But you have to start somewhere. The base terminology of a theory, such as geometry, cannot be defined in a normal fashion. Instead, we call these terms "primitives", and choose certain statements, called "axioms" or "postulates" relating them to each other which we simply assert to be true. It is this list of primitives and axioms that establish what the theory we are working on is. In a sense, the axioms define the primitives, by establishing how they relate to each other.
While it is common to treat "line" and "plane" as primitives, it is not absolutely necessary. You can build up all of Euclidean geometry just from the primitive term "point" and the primitive relationship "these two points are closer together than those two points", and basic set theory. (It is not wise to do so, as it requires a lot of niggling axioms to make sure that these primitives encompass the desired behavior, which would thoroughly confuse new students of the subject.)
The general outline of this approach is:
With this system, it is by definition that once a line and a plane share two points, the line must lie in the plane, as every point on the line must be colinear with those two points, and therefore must be in the plane, as the plane is linearly closed.
Whether you choose to do some grandiose scheme of definitions as above, or take the simple route of having "line" and "plane" as primitives and make this condition an axiom, it is something you want to have in your geometry, because it expresses the idea of what we want a plane to be: something that expands the idea of "straightness" into another direction.
Equation of line: $\frac{x-a}{l}=\frac{y-b}{m}=\frac{z-c}{n}$ ...(1)
Equation of plane: $\alpha x+\beta y+\gamma z+\partial=0$...(2)
For intersection, any point on line $P(tl+a,tm+b,tn+c)$ must lie on plane too. Plugging this point in equation of plane will give a linear equation in $t$ which may give exactly one $t\in\mathbb R$ (intersection is a single point) OR an identity $0=0$ true for all $t\in \mathbb R$ (intersection is a line).
A plane is a convex set. A convex (i.e. linear) combination of any two points in a convex set will be inside the set. So if you have two distinct points in the plane, you automatically have infinite points inside the plane.
Join the two points on the plane: that's a straight line which we'll call $\;L\;$ and which , btw, is wholly contained in the plane . If another line, say $\;M\;$, intersects those two points on the plane then it intersects $\;L\;$ in those two points. That cannot be, of course, unless $\;L=M\;$ ....
Think of it this way: We define a line-segment to be the shortest path in space between two points, and a straight line is the union of all line segments between any two of its own points. Then if a straight line intersected a plane in only two points, the segment between those points in the plane would be shorter than the segment between those two points in the line, contradicting the assumption that it is in fact a straight line.
The difficulty in proving this comes from the fact that whether or not a line, not on a plane, can intersect the plane in more than one place is equivalent to Euclid’s 5th postulate.
So in geometries where two parallel lines may intersect more than once, a line may also pass through a plane more than once.
Example: Consider two lines in a double elliptic geometry such as spherical geometry such that they intersect twice. If you place a plane such that one of the lines lies in the plane, then the other line will pass through the plane twice also.
If points A and B are in the intersection, then the midpoint of the line segment AB is also in the intersection, making three points (assuming A and B are distinct points). This can be continued infinitely, showing that the number of points in the intersection is infinite.
First, some definitions:
This leads to three relations between a line and a plane:
To make a line intersect the same plane twice it can't be parallel, and either the line or the plane must be curved.
