Is it grammatical to split either/or into different sentences?

I came across the following sentence in Wikipedia:

The bitangent lines can be constructed either by constructing the homothetic centers, as described at that article, and then constructing the tangent lines through the homothetic center that is tangent to one circle, by one of the methods described above. The resulting line will then be tangent to the other circle as well. Alternatively, the tangent lines and tangent points can be constructed more directly, as detailed below.

It took me a while to parse it because I was hopelessly looking for the missing or. Leaving alone the fact that this passage is clumsy, is it grammatical according to the rules of standard English?

In my view it certainly isn't.

The first sentence doesn't make sense because, as you point out, it uses 'either' without 'or'.

In order to make the whole piece grammatical the writer simply needs to leave out the 'either' (except they may have too many 'constructed/constructings').

These sentences make me cringe. I agree that "either" needs to be taken out to help it make some sense, otherwise the reader will just keep going through all of the awful comma splices trying to find the second part of that thought.

Usually Wikipedia has decent grammar. Just always keep in mind that all of the information on there is added by random internet-browsers like yourself, so don't take it too seriously.

Those sentences are so long, and so bursting with technical description, that the most sensible way of handling this much material is to break it all up into more manageable (and more comprehensible) segments.

If I were to rewrite the text you quoted, I would start off by explicitly stating that I was about to propose two different solutions -- perhaps with something like this,

There are two different ways to construct the bitangent lines.

followed by

The first method is to...

and then

The second method is to...

This way, the reader does not have to remember that he should be on the lookout for a clause beginning with "or" because he had spotted an "either" in the same sentence half an hour earlier.

I agree with the other answers, and will go further to say that it is incorrect to leave an "either" in that context without addressing an alternative in the same sentence.

It seems to me like the person meant to write an either/or sentence, and then got so carried away cramming that many words into it that he forgot the "or." Then, being lazy, he tacked on the "alternatively" sentence to make it seem passable.