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Of course the actual line in question is infinite in length so there is no mathematical reason to stop at any particular endpoints. So I don't think it would be wrong to have it extend a bit below the baseline (as long as the top endpoint also crept down a bit so that the length matched the parallel side. It seems simplest though to just end at the baseline.

But if the line fails to cross the grid lines at certain precise locations, that would be incorrect.

As for the circle, it is required that it pass through the centre of each O so it can't be tangent. If yours is tangent, it doesn't go through the centres.

Hi buzzb. What about the two Os in the bottom line? Surely their centres are in line with centres of the Xs which form the base line?

I think the time has come when some of these puzzles should come with a health warning. I retired early yesterday with a violent headache. It took ages to finally get to sleep. However, this morning I came back to the grid and the forum, and found, to my surprise, that I already had the line drawn, albeit too short. Reading your comments and extending the line revealed the name. So, thanks to you all. Headache is now gone.

merenz - the centres of the two Os at the bottom would be joined by a little arc of the circle which would lie slightly below the base line, so it wouldn't be a tangent ( but a secant.)
However if this is all being drawn on a grid the size of the one in the newspaper it is unlikely you would see the difference (as was pointed out previously.) i don't submit so it doesn't really matter to me but if you ask me (I know, you didn't) we are forgetting this is a crossword and not a mathematical treatise.

Thanks Gem for the helpful clarification. You’re right, we should not lose sight that it is a crossword. But I do love a good mathematical treatise :-)

I agree gems.

I’ve had another look at this this morning in the cold light of day, and read up on the nine point circle and the orthocentre. I have now concluded this is a fantastically clever construction.

The setter has managed to find a triangle with vertices in the middle of squares, where the orthocenter and every point on the 9 point circle is exactly in the middle of a square, and where the centre of the nine point circle is on an exact corner. In my view that is close to astonishing.

Buzzb, an earlier post of mine (#45) made clear that the circle in my solution is not tangent, and it is visibly not so because I have drawn thin lines. But as i pointed out, and as gem94 reiterates below, the use of a broad ballpoint or slightly blunt pencil might obscure the difference.

And PS, buzzb, thanks for responding to my question about location of the end points of the line. My initial solution had it starting on the base of the triangle, but I think I will move it down a fraction.

I do not know how picky John Green is going to be so I have no idea whether a circle that 'looks tangent' will be marked wrong. Clearly solvers are not going to be expected to draw perfect circles or perfectly straight lines. As soon as you give a line or circle any thickness at all you don't have a true line or circle in mathematical sense. My initial post was in reaction to an earlier in which someone said their circle was tangent and was meant as a friendly warning that this was incorrect (as is an Euler line that passes thru the centres of the EULER cells)

I am still struggling with filling the grid and I haven’t yet progressed to the mathematics. Any help would be appreciated with the following two clues:-

Life saving provided on a particular river down under

International sailors gathering over stern

Also I can’t parse the following two clues:-

Just about highest point orbiting round Earth?

The greatest Man U legend or worst