How did the mathematicians establish that parallel lines never intersect?

Answer 1

Definition of parallel lines: Lines that never intersect.
No autographs please.

Answer 2

When a mathematician asks a question, the first thing is always to make sure the meanings of all the terms are clear. So in this question, we have to first clarify what we mean by parallel lines.

Most non-mathematicians would probably say that parallel means always the same distance apart. If we take this as the definition, then obviously the definition implies that parallel lines can never intersect because they can never get closer together.

However, in mathematics, parallel lines are not defined by the distance between them. The definition given by Euclid over 2000 yrs ago is:

Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.

So again, with this definition, it is automatically true that parallel lines never intersect, that is, true by definition.

The answer to the question, then, is that mathematicians NEVER PROVED that parallel lines don't intersect. There was no need to, because parallel MEANS not intersecting.

Answer 3

It is easy enough to prove from the definition of parallel.

Let's say we have two lines that intersect. If you choose a random point on each line, and a third point where the lines intersect, you will have two vectors (using the word loosely). The vectors are parallel (or anti-parallel) if their cross product is zero, which occurs only when one or both of the vector magnitudes is zero, or when they are parallel. Since the vectors are non-zero, their cross product is only zero when they are parallel.

If they are parallel, then the angle formed by the vector 'wedge' must be zero, which means the lines are co-linear (they're the same line).

So, you cannot have two parallel lines that simultaneously
1) intersect
and
2) are not the same line

Answer 4

That's a very good question. In fact, mathematicians never did establish that parallel lines never intersect. The ancient Greeks who developed Euclidean geometry used it as one of their axioms (i.e., a "given"), but they never proved it.

Much later, geometers wondered what would happen if it was assumed that parallel lines DID intersect, and lo and behold, a whole new geometry (non-Euclidean geometry) was developed. In fact, there are a number of such geometries, and all of them are as "correct" as Euclidean geometry. They just stand on a different set of axioms.

Answer 5

Mathematicians have NOT established that.

Answer 6

umm common sense