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0|0|0|||||Impossible to find circle center with only a straightedge%3F|bennettjw||11:57:17|04/08/2010|Two questions%3A%0D%0A%0D%0AExactly how is the given circle identified without identifying a center point or a point whose distance from the center point represents the circle radius%3F  This makes little sense either theoretically or mechanically.  At least%2C my compass can not do it.  Maybe there needs to be a new postulate to Euclid%27s elements that every circle has a center point.%0D%0A%0D%0ADoes not the %22transformation%22 of the circle %28in three dimensions%29 represent a %22redrawing%22 of the circle%3B and hence%2C breaks the rule of only using a straightedge%3F  If not%2C then it implies a mechanical transformation%2C which invalidates the anti-proof.%0D%0A%0D%0A
1|1|0|||||RE%3A Impossible to find circle center with a straightedge%3F|alexb||12:13:37|04/08/2010|%3EExactly how is the given circle identified without %0D%0A%3Eidentifying a center point or a point whose distance from %0D%0A%3Ethe center point represents the circle radius%3F  %0D%0A%0D%0AA circle may have been drawn using a stencil%2C or in the usual manner - by somebody else%27s anscetor - so long ago that the paper%2C like an old map%2C grew so decayed and fragile that while the lucky adventurer tried to smoothen it out on a table%2C the middle part caved in and disintegrated.%0D%0A%0D%0AHave you seen a metaphorical interpretation of Bottema%27s theorem%3A%0D%0A%0D%0Ahttp%3A%2F%2Fwww.cut-the-knot.org%2FCurriculum%2FGeometry%2FBottema.shtml%0D%0A%0D%0A%3EThis makes %0D%0A%3Elittle sense either theoretically or mechanically.%0D%0A%0D%0AWhy%2C it did a lot of sense to several generations of geometers starting in the mid 1800s.%0D%0A%0D%0A%3EAt least%2C my compass can not do it. %0D%0A%0D%0AIt certainly takes imagination rather than brute force.%0D%0A%0D%0A%3EMaybe there needs to be a %0D%0A%3Enew postulate to Euclid%27s elements that every circle has a %0D%0A%3Ecenter point. %0D%0A%0D%0ABy all means. You should write to somebody responsible for improving on Euclid%27s axioms.%0D%0A%0D%0A%3EDoes not the %22transformation%22 of the circle %28in three %0D%0A%3Edimensions%29 represent a %22redrawing%22 of the circle%3B %0D%0A%0D%0AI do not know. It all might be in the mind%27s eye. How do you draw a circle in 3d%3F%0D%0A%0D%0A
2|2|1|   ||||RE%3A Impossible to find circle center with a straightedge%3F|bennettjw||12:41:59|04/12/2010|%3E A circle may have been drawn using a stencil%2C or in the usual%0D%0A%3E manner - by somebody else%27s anscetor - so long ago that the paper%2C %0D%0A%3E like an old map%2C grew so decayed and fragile that while the lucky %0D%0A%3E adventurer tried to smoothen it out on a table%2C the middle part %0D%0A%3E caved in and disintegrated.%0D%0A%0D%0AThe explanation of stencil or ancestor still begs the question%3A How do you know it is a circle%3F  Keep in mind that Geometry is not about the %28mechanical%29 drawing%3B but rather%2C it is about the mathematically exact line%28s%29 or circle%28s%29 it represents %28theory%29.  We know that when we draw a circle or arc with a compass that we are representing a theoretically exact circle because we are representing every point on the circle edge is being equidistant from the circle center point.%0D%0A%0D%0APerhaps a better way to present the problem is %22Is it possible to find the circle center point without using the circle center point and only using a straightedge.%22%0D%0A%0D%0AMy whole issue with this problem and its proof is that with only an existing circle and straightedge one can only draw random lines.  One can say when a line intersects the circle%2C and one can draw more random lines through those circle intersection points%2C but one can do or say nothing else to include when the circle center has been found at some time in the infinite future. In short%2C drawing circles enable some ability to find direction.  If drawing circles are removed from the set of tools%2C then not just this problem%2C but any Geometry problem is reduced to nothing of interest or value.%0D%0A%0D%0A%22Anybody find a circle center point around here%3F  I seemed to have lost one.%22%0D%0A%0D%0A%3E Why%2C it did a lot of sense to several generations of geometers %0D%0A%3E starting in the mid 1800s.%0D%0A%0D%0AAmateur geometers or Professional %28w%2F degree%29 Geometers%3F Recall that Wenzel discovered in 1837 an anti-proof that such problems as Angle Trisection could not be done%2C which heralded in a time of great interest in understanding Greek Philosophy and Mathematics by a lot of people%2C a vast number of who were amateurs.  This might have been of great interest to amateurs%2C but I doubt this was of any interest to professionals.%0D%0A%0D%0A%3E It certainly takes imagination rather than brute force.%0D%0A%0D%0AImagination without reason does nothing to %5Bpreserve the best of Geometry %28Hippocrates%29%5D.%0D%0A%0D%0A%3E How do you draw a circle in 3d%3F%0D%0A%0D%0AIt requires identifying the plane that the circle lies in.  Identifying a plane requires the following 3d conditions%3A%0D%0A- Two intersecting lines%2C%0D%0A- Two parallel lines that are each not the other%2C%0D%0A- A line and a point not on the line%2C%0D%0A- Three points that are each not the other%2C%0D%0A- A circle%2C or%0D%0A- Any combination of lines%2C points%2C or circles whose subset creates the above 3d conditions %28e.g. two intersecting circles forming a line%29.%0D%0A%0D%0AThis last was actually a more interesting question than the topic problem.%0D%0A%0D%0AI guess the larger issue is not the problem%2C but the acceptability of the proof that it can not be done.  Let%27s try to form the frame of a more reasonable proof this way.%0D%0A%0D%0AFirst%2C start by working only in the plane of the circle.%0D%0A%281%29 Draw a line. This can be through%3A%0D%0A   - the circle%3A %0D%0A      - through a previous intersecting point formed by a previous %0D%0A          line%2C or%0D%0A      - through two previous intersecting points formed by two %0D%0A          previous lines%2C or %0D%0A      - through no previous intersecting points%3B%0D%0A   - not through the circle. %28Only intersections with the circle can %0D%0A      possibly provide information regarding the circle center.%29%0D%0A%282%29 Does the line intersect the circle%3F%0D%0A   a%29 No%2C go back to step %281%29. %0D%0A   b%29 Yes%2C continue.%0D%0A%283%29 Does the line intersect the circle center%3F%0D%0A   a%29 Since no information can be derived from the two intersecting %0D%0A      points of this line with the circle%2C then the answer is%3A Don%27t %0D%0A      know. Go back to step %281%29.%0D%0A%0D%0ASecond%2C does considering an additional dimension add any more information regarding the circle center regarding any line we can draw%2C intersecting or not%3F  No. Done.%0D%0A%0D%0A%0D%0A%0D%0A%0D%0A
4|3|2|||||RE%3A Impossible to find circle center with a straightedge%3F|alexb||12:52:33|04/12/2010|%3EThe explanation of stencil or ancestor still begs the %0D%0A%3Equestion%3A How do you know it is a circle%3F %0D%0A%0D%0AExactly same way that you know that a straight line can be darwn with a straightedge.%0D%0A%0D%0A Keep in mind that %0D%0A%3EGeometry is not about the %28mechanical%29 drawing%3B but rather%2C %0D%0A%3Eit is about the mathematically exact line%28s%29 or circle%28s%29 it %0D%0A%3Erepresents %28theory%29.  %0D%0A%0D%0AAbsolutely%2C I am sure you have imagined a straightedge. Now try imagining a curcular stencil.%0D%0A%0D%0A%3EWe know that when we draw a circle or %0D%0A%3Earc with a compass that we are representing a theoretically %0D%0A%3Eexact circle because we are representing every point on the %0D%0A%3Ecircle edge is being equidistant from the circle center %0D%0A%3Epoint. %0D%0A%0D%0AYes%2C this is a common convention.%0D%0A%0D%0A%3EPerhaps a better way to present the problem is %22Is it %0D%0A%3Epossible to find the circle center point without using the %0D%0A%3Ecircle center point and only using a straightedge.%22 %0D%0A%0D%0AWhatever does it for you.%0D%0A%0D%0A%3EMy whole issue with this problem and its proof is that with %0D%0A%3Eonly an existing circle and straightedge one can only draw %0D%0A%3Erandom lines.  %0D%0A%0D%0AThis a sweeping claim. One may try drawing the lines purposefully%2C with a certain idea%2Fgoal in mind.%0D%0A%0D%0A%3E%3E Why%2C it did a lot of sense to several generations of geometers %0D%0A%3E%3E starting in the mid 1800s.%0D%0A%3E%0D%0A%3EAmateur geometers or Professional %28w%2F degree%29 Geometers%3F %0D%0A%0D%0AProbably amateurs too but I know only about the professional geometers because of the literature they left.%0D%0A%0D%0A%3EThis might have been of great %0D%0A%3Einterest to amateurs%2C but I doubt this was of any interest %0D%0A%3Eto professionals. %0D%0A%0D%0AIf you look into 100 Great Problems of Elementary Mathematics by H. Dorrie%2C you%27ll see that %2334 is call %22Steiner%27s Straightedge Constructions%22. The theorem at hand is also Steiner%27s%2C I believe.%0D%0A%0D%0A%3E%3E It certainly takes imagination rather than brute force.%0D%0A%3E%0D%0A%3EImagination without reason does nothing to %5Bpreserve the %0D%0A%3Ebest of Geometry %28Hippocrates%29%5D.%0D%0A%0D%0AWhat about the brute force%3F%0D%0A%0D%0A
3|2|1|||||RE%3A Impossible to find circle center with a straightedge%3F|bennettjw||12:41:59|04/12/2010|%3E%3E Imagination without reason does nothing to  .%0D%0A%0D%0AShould read%3A%0D%0A%0D%0AImagination without reason does nothing to %22preserve the best of Geometry %28Hippocrates%29%22.%0D%0A%0D%0A