I wondered which spelling was correct after recently seeing each one.

Problem:
Each spelling is considered to be correct, but not everyone agrees which among these three spellings is/are correct.

Explanation:
These three words mean the same thing as nouns, they mean the same thing as verbs, and they mean the same thing as adjectives.

The different spellings here represent historical variance.

For fun, I searched Google for each of the following words (with the quotation marks, to avoid variations) and got about the indicated numbers of matches:

• “coordinate” — spelled C-O-O-R-D-I-N-A-T-E — 52,800,000 matches
• “co-ordinate” — spelled C-O-HYPHEN-O-R-D-I-N-A-T-E — 5,050,000 matches
• “coördinate” — spelled C-O-DIAERESIS-O-R-D-I-N-A-T-E — 10,100 matches

This tells me that Web authors have used these three spellings by a compound ratio of 5,228-to-500-to-1.

This compound ratio seems to reflect the evolution of the word. From what I have seen, “coördinate” with an diaeresis-O was the original spelling of the word. This seems to have morphed into “co-ordinate” with no diaeresis but with a hyphen between the first “o” and the second “o”, which seems to have morphed into “coordinate” with no diaeresis and no hyphen.

I believe that evolution of this word is consistent with my “Devolution toward Simpler” linguistic hypothesis. It is simpler to write or type a hyphen than to write or type an diaeresis, and it is simpler to omit the hyphen than to include it.

Solution:
Use any these three spellings of the word, but realize that the most popular spelling today has no hyphen and no diaeresis.