^{1}

A set of reals

Let us say that a set of three real numbers

A

If

Alternatively, any set

In contrast with the usual arithmetic viewpoint, the geometric viewpoint adopted here takes us in an apparently novel direction, where the sets of interest are most naturally viewed as subsets of

In what follows, a key tool for discussing midpoint triples will be base 3 representations of the real numbers, so relevant notational conventions will now be specified. If

Since

These conventions are extended to

If

The regular base 3 representation of any nonnegative rational

Base 3 representation of negative reals is simply achieved by writing any such number as

To distinguish between instances of base 3 and base 10 representations, an explicit subscript 3 is normally attached to the former, while a subscript 10 is normally kept implicit for the latter. As needed,

Let

Some features are obvious. Clearly

It turns out that these three sets are of considerable interest when it comes to the presence or absence of midpoint triples, so let us now address the main subject of this paper.

Note that the set

Not only is every member of

In contrast, it turns out that

On the contrary, suppose that

It will now be shown that

Given

It will now be shown that extending

Again we seek

For instance, the proofs of Claims B and C produce the decompositions

The sets

Let us now study

Calculations until now have involved sums of pairs

Given any positive integer

If

If

If

For

By doubling, it follows from Claim D that, for any integer

The two sets forming this union are certainly midpoint-free, so any midpoint triple in the union must have two members in one set and one member in the other. Claim E shows that there is no midpoint triple with exactly one member in

Since

The sets

Now let us consider the corresponding subsets

As expected,

Next, consider

If

We seek

In particular, if

Positive rationals not in

We seek

If

The sets

Now consider

Suppose

Given

By Claims G and J, if the specified set contains a midpoint triple, two members of the triple must have one sign and the third member must have the opposite sign. Suppose

Given

Now suppose

Finally, suppose

For instance, beginning with

It now follows from Claims I, K, L, and M that the set

What is the situation for

If

If

The preceding results now fully establish the following.

The two sets

Now consider

The sets

The two sets

The author declares that there is no conflict of interests regarding the publication of this paper.