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Given a circle $O$ on a paper, we do not know the centre point. Can we draw the centre only using a ruler (by which we can only draw straight lines)?

One fact I know: we can draw the tangent line at a point on the circle, by Pascal theorem.( or Noether's Max Fundamental theorem )

Thanks.

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How can we use Pascal's theorem to construct a tangent? –  mixedmath Aug 9 '11 at 7:20
    
Let the hexagon inscribed in the conic become pentagon. –  wxu Aug 9 '11 at 7:24

3 Answers 3

up vote 10 down vote accepted

According to wikipedia the answer is no:

In geometry, the Poncelet–Steiner theorem on compass and straightedge construction states that whatever can be constructed by straightedge and compass together can be constructed by straightedge alone, if given a single circle and its centre. This result is the best possible; if the centre of the circle is not given, it cannot be constructed by a straightedge alone.

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What about the tangent line? We are not given the centre but we can construct the tangent line. –  wxu Aug 9 '11 at 7:16
    
@wxu, are you asking a question? You say that you know you can draw a tangent line, and you say you know how, so what are you asking? –  Gerry Myerson Aug 9 '11 at 9:35
    
:I donot know why the Poncelet–Steiner theorem indicates that the answer is no. I have some troubles to understand what the theorem says. Does it indicate that: if given a unit length 1, we can construct $\sqrt{2}$ by straightedge and compass; so by this theorem, we can construct $\sqrt{2}$ too only using straightedge alone and any given fixed single circle and its centre? –  wxu Aug 9 '11 at 14:04
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@wxu, yes, I think that given straightedge, one circle, center of that circle, and 1, you can construct $\sqrt2$. See also cut-the-knot.org/impossible/straightedge.shtml for a proof that it's impossible to find the center of a given circle with straightedge alone. –  Gerry Myerson Aug 10 '11 at 0:47
    
Thank you very much. If so, the statement of this theorem indicates the answer is no. Amazing theorem and interesting link. thanks. –  wxu Aug 10 '11 at 4:35

Ok - it's too long to comment. I speak of possible additions to make it possible, as Gerry has indicated that it's not.

There is a big list of really small things that would make it possible. Having a ruler, rather than a straightedge would work. Or a straightedge with only a few marks (if allowed to make them ourselves beforehand - just 2 or 3, depending on whether the ruler is finite or not, to make right angles). Or the ability to make right angles. Alternatively, the ability to draw a line parallel to a given line (although this is cheap, as it's equivalent to finding midpoints of line segments). Or an inscribed parallelogram (really, we need two secants to the circle that are not parallel to eachother, and one line parallel to each to do this easily). If we don't mind building up a sweat, a circle and a parallelogram (anywhere) will do the trick too.

Or one secant and a parallel, and either one mark anywhere on the ruler. If we can make 1 mark anywhere on the ruler, then we don't even need the secant or its parallel (to distinguish - if the ruler comes with a mark on it, we need more. If we can place a mark on it, i.e. we can match one given length, we don't need anything more). Or another circle that intersects this circle in two places. Or a piece of string of fixed length (able to trace an arc of one particular curvature).

There are many more, I'm sure. But what I'm trying to get at is that this seems to be a good cusp - almost any additional information is enough information.

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We can throw away the straightedge and fold the paper. –  André Nicolas Aug 9 '11 at 11:46

Make a line intersecting the circle in two points A, B. Now construct the perpendiculars to the line at A and at B. These lines meet the circle again at C, D respectively and the lines BC and AD meet at the center. This works as along as the starting line is not a diameter.

So if you can make perpendiculars you can construct the center of the circle. But you can't make perpendiculars just from line constructions.

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Maybe it is a matter of definition, but I think you cannot trace perpendiculars with a straightedge. –  alfC Sep 28 '12 at 5:35
    
@alfC It needs to be proved. Assume you can do perps then you can find center of circle as I show, and then you can do ruler-compass constructions..... –  i. m. soloveichik Sep 29 '12 at 13:53
    
In my definition of straightedge-and-compass (also I think the usual one), you need a compass to draw the perpendicular to a line. The compass gives you the "metric" in space as it allows you to rotate points at fixed distances. –  alfC Oct 2 '12 at 21:57

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