Greetings, gentle patron!
If you are encountering difficulty making progress on my ninth ænigma, perhaps you'll find relief in one or both of the following pedagogical aids.
First, I have collected a small assortment of advisory notes, or ‘tips’ as one might say, which are enumerated a bit below on this page.
I trust that these resources will suffice to lift your mental ship off the shoals of perplexity and allow you happily to continue your journey down the river of clarity and decipherment, but if you find that they do not, I invite you to contact my associate, Pavel Curtis, directly for more individualized furtherance.
With my very best wishes for your imminent enlightenment,

As with so many ænigmas, it is ever so much easier to proceed when one has as much contextual information as possible. In this case, perhaps the most important information you can record is every segment of region boundary that you can deduce. Each time that you can definitively associate a square with the dot to whose region that square belongs, you may copy any boundary segments around that square to the rotationally symmetric square within the region. Pray do not hesitate to draw every such boundary segment immediately, as soon as you discover them.

Right at the beginning, there are many such boundarysegment copying opportunities available to you, arising from two different sources: dots occupying squares adjacent to the edge of the grid, and dots occupying squares adjacent to squares occupied by other dots. These sources provide a quite substantial number of initial clues to get your solving experience under way. I encourage you to expend all of your early efforts on the propagation of these starting points. A perhaps surprising amount of the grid will thereby “solve itself”, right before your (no doubt) grateful eyes!

When no further such simple boundarycopying steps are available, you will be forced to explore other deductive avenues. Your most potent tool at these times will be finding squares that could only be legally associated with a single dot's region. Choose a square that seems likely to you (your intuitions about such things will, I am certain, quite quickly mature), and consider the question, "with which reachable dots could this square possibly be associated?"
A square can only be associated with a dot if (a) there is a path, from that square to the dot, that passes through no other region, and (b) the rotationally symmetric path (180 degrees around the given dot) also passes through no other region.
When you find that a square can only be associated with a single dot, I recommend that you lightly draw in the path from the square to the dot, along with the rotationally symmetric path, thereby visually reminding yourself of the substance of this deduction (and possibly also making clear the locations of some new boundary segments to be drawn). Do be careful, though: sometimes there will be several possible paths between the square and the dot; you should only notate those parts of the path that are in common among all of the possibilities.

As always with my ænigmas, progress toward the solution here will come almost entirely in the form of small, incremental steps, often adding but a singleunit boundary segment or a unitary square of known region membership to the grid. Do not endeavour to rush ahead, attempting to guess or otherwise ascertain an entire region shape in one fell swoop. The whole of your solution will come as the sum of many steps; “patience” and “perseverence” must be your watchwords.
Wishing you the best of solving luck,

