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Despite my despair at creating a Connect Fano Plane in the spirit of a normal Connect Wall, I think I succeeded, thanks to the database of world cities I downloaded from Geo Data Source. In this iteration, you are given seven municipalities from around the world. Your job is to determine seven categories such that the municipalities and the categories form a Fano plane, as in this diagram:

Fano

For those of you who did not see the previous iteration, the rules are:

  1. The municipalities are the points, and the categories are the lines (the circle is a "line").
  2. A municipality can only be in the three categories corresponding to adjacent lines, and must specifically not match the other categories.

If you get the categories correct, there is a final connection (spoiler: it's a municipality) that matches all seven categories. A complete answer consists of the Fano diagram with the municipalities placed, a description of the seven categories, and the final connection. Ready?

ALEXANDRIA, ANGOCHAGUA, HUAZHOU, NEGAGE, NOUAKCHOTT, OTTAWA, PYONGYANG

Note: ALEXANDRIA refers specifically to the largest city in the world with that name.

As an early hint:

Three of the categories decribe geopolitical properties of the municipalites; four are language properties of their names (as transliterated here).

GOOD LUCK!

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How about this arrangement:

enter image description here

NOUAKCHOTT, OTTAWA, PYONGYANG – capital cities (of Mauritania, Canada and North Korea, respectively);

ALEXANDRIA, NEGAGE, NOUAKCHOTT – found in Africa (in Egypt, Angola and Mauritania, respectively);

HUAZHOU, PYONGYANG, ALEXANDRIA – population greater than 1 million (1.1m, 2.9m and 5.2m, respectively);

OTTAWA, ALEXANDRIA, ANGOCHAGUA – contains 2 or more 'A's;

ANGOCHAGUA, NEGAGE, PYONGYANG – contains 2 'G's;

HUAZHOU, ANGOCHAGUA, NOUAKCHOTT – contains the letter 'U';

HUAZHOU, OTTAWA, NEGAGE – contains exactly 3 consonants.

Note that the populations of Nouakchott and Ottawa are currently estimated to be in the 900,000s and very near 1 million, but officially they're not quite there yet!

If all of these categories are correct, then the municipality which would satisfy all 7 categories is:

OUAGADOUGOU:
- Capital city of Burkina Faso;
- In Africa;
- Population 2.2 million;
- 2 A's;
- 2 G's;
- 3 U's;
- 3 consonants (G, D, G).

PS It's also worth noting from the comments that there are a few alternative linguistic categories that would also work to produce the same final connecting municipality...

- Category 4 ("Contains 2 or more A's") could equally be "Begins and ends with a vowel" (@JeremyDover - OP);
- Category 6 ("Contains U") could be further constrained as either "Contains the two-letter sequence 'UA'" (@M.Boligsou) or "Contains A, O and U, and no other vowels" (@JeremyDover - OP);
- Category 7 ("Contains exactly 3 consonants") could be further constrained as "Contains exactly 3 consonants, 2 of which are the same" (@M.Boligsou).

Replacing one of my chosen categories with one of these would result in the same final answer.

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    $\begingroup$ Most of the way there, but do you have the final connection? $\endgroup$ – Jeremy Dover Jul 10 at 20:32
  • $\begingroup$ @JeremyDover I think so! Hit 'Post' prematurely before writing up that part! :) $\endgroup$ – Stiv Jul 10 at 20:34
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    $\begingroup$ @JeremyDover By the way, I think this approach works even better than the one in the first puzzle. And the final full-7 connection gives you a way to check your answer. I think you've developed a very cool new puzzle type here, although I imagine it will be very difficult to run a full series of them! Must have taken you a while to put together. Well done :) $\endgroup$ – Stiv Jul 10 at 20:38
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    $\begingroup$ You just beat me! A few of my categories were a bit more specific; rot13(nyy gur zhavpvcnyvgvrf va pngrtbel 6 pbagnva HN naq nyy gur zhavpvcnyvgvrf va pngrtbel 7 unir guerr pbafbanagf ohg bayl gjb havdhr). Nice job! $\endgroup$ – M. Boligsou Jul 10 at 20:38
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    $\begingroup$ Thanks for the feedback...I think I prefer this too. Actually only took me about 90 minutes to put together, but I had a very broad pool of locations to choose from. Even with that, finding the seventh word which had to rot13(ortva naq raq jvgu n ibjry; unir NBH ohg ab V be R; naq unir gjb Tf) was tough. But then as you showed, I made it more complicated than I needed to. I think I've realized that with only four words not in a category, I don't have enough degrees of freedom to make the language categories complicated. Oh well...glad you enjoyed, and I'll try to put another together! $\endgroup$ – Jeremy Dover Jul 10 at 20:46

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