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Comment by bee_rider | Hacker News Reader

It seems to me that the placing of one of the points is “exact.” Unless you are working on graph paper or something (but that sort of goes against the spirit of classic geometry I think), there’s no absolute frame in which the first point can be incorrect, the rest of the construction is relative to it, right?

Also, is the inaccuracy around B actually a circle? I think you could easily end up missing that horizontal line he’s added between the imagined “right A” and “right B.” But, that line shouldn’t exist yet, it connects A and B wherever they are. The position of B is wherever your pencil lands.

The second circle seems to have more room for error. You can miss B with the pointy end, and could flex the compass differently from when you did the first circle, resulting in a different BA distance. Although, it should be possible to visually check this.

Is that right? I somehow never managed to take a geometry class.

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I think the circles are for when you create the arc using the compass.

Presumably you already constructed points A and B and joined them into a segment, so there's now a point B to miss.

If you are just trying to show the relationship between the lines, I don’t think there’s any reason to set A and B beforehand.

Of course, it could be the case that there’s a multi-step proof where you do care about the positions of A and B in relation to some other things. But I think that would require doing the analysis with that in mind. Maybe the order in which you construct the more complicated proof can also benefit from picking certain points to be exact.

I think the construction is usually phrased: "given a line, to..", so it assumes a pre-existing line segment, rather then that being part of the construction itself.

A line is defined by a pair of points, isn’t it? It is literally the line AB, wherever A and B are, they define the line.

But you start the construction with the line already drawn. Then you have to place the arms of the compass on the points, so you have error on both ends.

I must be missing something; I don’t see how you can rigorously define the line without A and B, so I don’t see how it can exist before A and B are marked.

I think it is possible that part of what’s missing in your “given a line to..” is that the description of the line in the .. refers to A and B, or possibility the phrasing is just informal.

A, B and the line connecting them exist before you start the construction, and the author assumes there's no error there.

Then to start the construction, you have to put one arm of your compass on the point A, and one on the point B. These points exist and are already marked, but you incur error when you place your compass down on each one. These are the represented by the blue and yellow circles in the diagram.

> I don’t see how you can rigorously define the line without A and B

That line segment isn’t defined, it’s postulated (“there somehow is a line segment with endpoints A and B; you needn’t care how it came about”)

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