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I'm with Beepy, I think.

The usual letter for "base" is E. Put that does not account for (compass) "point" in the clue, does it?

Hi, woodlouse.

I parsed this as "prominent star" SOL + E for "bottom point" and the BRB gives "exclusive" as a definition for SOLE.

Beepy is right, I think. The wordplay is "point after prominent star". That is sandwiched between two definitions. I think I have come across such a clue before, but it's very rare. I actually quite like it since the wordplay gives the solver something to work from. Double definition clues can be difficult solve until there are some letters in the grid, and they are also rather dull clues in most cases.

I'm persuaded

But in that case I don't like the clue!!

Just to amend what I said earlier in response to buzzb's point that the two shapes are not tangential at any point: that is just about apparent in an accurately drawn diagram in Photoshop with a line thickness of 2 pixels in a 72 ppi image. A hand-drawn solution is likely to have thicker lines that will blur the distinction. I doubt that the marker is going to be particular about that.

It doesn't touch. Look at the second across clue and you'll see the point.

Moffat, could you please explain that. What doesn't touch what?

If, in addition to everything else, we have to understand the impact that the de Longchamps point has on the solution, then this puzzle is far more complicated than I thought. I assumed that the only purpose of the circle is to provide a point in the grid that enables a solver to locate the Euler line precisely, given the need for parallelism. I've constructed all three intersections that give the Euler line, and everything works out as per the preamble.

Thanks for the feedback on my previous query.

I thought this was good, taught me something new. I also dislike the penultimate across clue (as discussed above) but I guess it’s OK. More problematical for me are the two unches in the top right and bottom left down clues, which is fairly unforgivable at the best of times, but for me is very poor in a carte Blanche. I was convinced my grid was wrong for a long time because of this.

I have about 7 clues that I can’t figure out the wordplay for but I’m fairly sure they’re all right, I guess I am not on the same wavelength as the setter.

Finally I assume the final step should go through the x rather than the centre of the guide cells, is that right?

I thought this was a fantastic puzzle and am pleased that my self imposed rule of not looking at this site until I’m done, remains unbroken. However, I do accept the endgame would’ve been something of a challenge to the non-mathematically minded.

I didn’t spot the Os initially, rather I discovered them having drawn the triangle on tracing paper, then used compasses and ruler to construct the circumcentre, orthocentre and the feet of the altitudes... the centre of the 9 point circle I found with a bit of trial and error. When I laid this over the grid I spotted the Os. My triangle’s base does indeed appear tangential to the circle but I appreciate this might not be the case if it was drawn on a larger scale and with super accuracy.

I note that some folk have mentioned drawing the line going through one square above the centre of the circle but this surely isn’t right? I can see why it’s appealing as it’d mean the line goes through the centre of the squares of the name. But then it misses the centre of the 9 point circle which, by definition, must also be on the line. And whilst the de Longchamps clue was a clever link to the theme, I don’t think it’s pertinent to the final construction.

It’d be reassuring to hear that the distance between my parallel lines is hopefully the same as others. I have it at 2.3cm