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isn't that clue the last one missing a letter in the wordplay ..the E?

That must be it Smellyharry, bete as in beast. I was misreading my own answer and looking for bite as in trying to get someone/thing on the hook as it were.
The contributor looking for Noel_ needs an e I think (haven't got the crossword here).

Kirky - thanks! I'd written down the 9 letters I needed and failed to allocate the last one.

I was on holiday in Greece until yesterday evening, so have only just come to this. At first, I was impressed that anyone could make a puzzle out of the geometrical configuration (one I know very well, as I have given masterclasses about it) and I looked forward to drawing the details. However, when I got around to this , I starting doubting the accuracy of the puzzle..

The preamble asks us to draw the Euler line precisely. I assume that the triangle in question has three Xs as vertices. Then it is straightforward to identify four significant points on the line. The centroid (usually denoted by G) is just to the the second E in A*****. The orthocentre (usually H) is the G in N*********. The circumcentre (usually O) is more or less the P of C*****. The nine-point centre (usually N) is at a grid point in the square consisting of the letters P,E,L,L reading anticlockwise (another mathematician); it is the midpoint of HO.

So I know precisely (as the instruction demands) where the Euler line is, provided that I have the triangle right. But this line is NOTparallel to a side of the triangle, and it does NOT pass through the letters of EULER which are equally spaced in the grid. However, the nine Os on the nine point circle are the midpoints of the sides, the feet of the altitudes, and the midpoints of the segments joining the orthocentre to the vertices. All of that matches up nicely, so I do not think that I have the wrong base triangle. But if I do represent the Euler line precisely, I will be marked wrong, as I am pretty sure that we are meant to join up the letters of Euler.

I can only think that I am wrong in assuming that the base triangle has Xs as vertices! But then everything else fits together so well.

(Incidentally, the nine point circle is not tangential to the side of the triangle, as someone was claiming: it never is. But that is irrelevant. However, I thought that Longchamps was a nice touch.)

Foinavon

Hi. Sounds like you are considerably more expert in this than me. I have a mathematical background but hadn’t come across the Euler line before. For what it’s worth I did some of these calculations and came to the conclusion that the orthocentre is exactly in the centre of the G, as you said. I think the centre of the nine point circle is at the exact corner of the cells E,L,P,E. That seems to me to be exactly root (12.5) squares from the centre of each of the nine O’s, and they seem to me to be exactly the nine points.

If that’s the case then the line joining those two points is the Euler line I think and I think It is parallel to one side of the triangle and it does go through the five guide cells.

Yes, it is ELPE - that was a misprint. I agree that the NPC is there and the ortho is at centre of G. But then the Euler line does not pass through the letters of EULER, so we are back where we started. Perhaps I have something wrong.

It doesn’t go through the middle of the EULER letters but it goes through the top left corner of each (and therefore through the left third of each).

My reading of it was that "guided by" was used to indicate that these letters would indicate which of the sides the line was to be parallel to, and the line would pass through the cells, but not necessarily through the centred.