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Fg, look at my last post.

The epicentre of the circle would appear to me to be at the point of the cross inside a box reading ELEP starting from northwest and going clockwise. This would slightly miss the line (which I spotted first and helped me confirm the other shape). The line would pass roughly a square higher.

Thanks unclued, yes, got there now but have to agree with sentiments already expressed that this has become too much of a mathematical puzzle for my liking!

Fg, what are you being asked to draw and who invented it? If you know that, find the inventor, equally spaced.

Your solution will be incorrect if:

a) the two shapes are tangent anywhere
b) the line is not PRECISELY positioned - no approximation is involved

The construction here is superb - all of the key letters are perfectly arranged. It's easy to miss what is going on with the O's.

Thanks. I can satisfy conditions A and B easily enough, but I still can’t see any justification in placing X at the centre from any of the diagrams I’ve seen. I can see a justification for placing O at the centre from some of the diagrams, presumably on the correct line, but precisely where along it escapes me.

Actually, having clicked on another link, I think I see what to put at the centre and how to complete the crossword. My final step wasn’t quite right originally, as I was applying a rule for entering the shapes to the whole solution, and I think that if I don’t do this then I can arrive at a solution which seems correct to me, at least.

Just one small thing niggling at me. The clue for the penultimate across entry appears to have the form DEFINITION + WORDPLAY + 2nd DEFINITION. I don't believe I have ever come across a clue with such a construction. Have I parsed this correctly, or am I way off?

I do not understand buzzb's assertion that a solution will be incorrect if the two shapes are tangent anywhere. Technically the circle crosses the base line of the triangle at the centres of two cells, namely the O's of the last across clue, therefore it is not tangential. But that would only be clear on a pretty large grid and using very thin lines . Surely on a submitted solution the bas line will look like a tangent of the circle.

Beepy - no double definition involved.

Bottom point is "base", using the usual single letter, after a word for "sun".