Whenever I draw a line through a circle on a surface, or the other way around - a circle over a line- the line moves! It stops being straight. I can understand drawing to/from points at the start of a segment on a circle, to force closed surfaces, but I do NOT want this.

How can I avoid it, making a segment fall at the right point, or at least forcing sketchup to orientate the circle rather than move my straight line! I know this would only ever work if the number of segments is a multiple of the arc length - can I make Sketchup do this?

You are seeing the default edge-breaking actions in SketchUp.
New geometry will automatically intersect with any geometry that already exists in the same context.
This is often the preferred outcome.
BUT if you want the circle NOT to intersect with existing geometry, then first make a group [or component] of the existing rectangle etc, then draw over that to make the circle, which will no longer be in the rectangleâs context and so it will not interact with it.

Later on if you wish to change the rectangle you simply edit the group [or component] and when working inside its context intersections will be reestablishedâŚ
Groups and Components are used to âseparate geometryâ, otherwise geometry will interact [usually by intersection or merging].
Do not think to use Layers to separate geometry.
Layers are used simply to control visibility.
Draw all geometry on Layer0 and assign other Layers to the Groups/Instances you make containing that geometry.
That way the interactions are prevented and the visibility is properly controlled, without unpleasant surprises down the lineâŚ

This splitting of edges should not move any vertices.
BUT if your geometry is very tiny, then the 1/1000" tolerance might be reached though ?
Scale up and see what happensâŚ

Alternatively, you can switch off the auto-intersect by using this code-snippet in the Ruby Console, applies during that session.

Sketchup.break_edges=false

To reinstate the default use:

Sketchup.break_edges=true

However, I recommend that you learn to use Groups/Components and get used to how SketchUp makes and interacts its geometryâŚ

What size is your model? I have seen this happen when the model is small enough that the line passes within SketchUpâs 0.001" vertex tolerance of an endpoint on the circle. SketchUp will âbendâ the line over to break at the circleâs vertex. If you are working at small size, you will need to scale the model up first, draw your geometry, then scale it back down to final size.

Thank you for your prompt and informative info. I was using dimensions around 0.1mm. I think for lines this was fine. The problem came when I was putting circles, which of course put points down at a much finer pitch than this. I think thats why I wasnât able to up my circles above ~50 segments also.

So - I am multiplying by 10 ) and will try the scale function at the end.