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New apparent porism  real?
Posted:
Jul 29, 2002 1:06 AM


(also to sci.math.research and geometry.puzzles)
This may or may not be correct but it's similar to Steiner's and Poncelet's porisms, and quite amazing at first glance. Take two circles, nonconcentric, one inside the other. Define a "model" triangle ABC with fixed angles. All triangles used in the chain are similar to this model. Draw one with two vertices, say A and C on the outer circle, C counter clockwise from A, and B on the inner circle. Draw another in the same position with its C on the previous A. Continue, head to tail in this manner, until you get to one that has its A close to the first C. Now adjust the radii or the center offset of the circles or the angles of all the triangles until the coincidence is extremely close to zero. For many combinations of these parameters, the reported error of closure is very low. For example, in the spirit of full disclosure of experimental data, if (all distances are in cm as drawn): Outer circle radius: 7.84717 Inner circle radius: 6.21058 Distance between circle centers: 1.33288 A, degrees: 29.709 B, degrees: 135.905 C, degrees: 14.386 Number of triangles for closure: 11 then the closure error, reported by Geometer's Sketchpad, varies from .00001 cm to .00062, which in percentage of the outer circle radius, is about 0.008 %. For fewer triangles or more offset between the circles, the error can get up to .12 cm or so, in the range of 1 or 2%. I think this error is real and not a measuring artifact, so I doubt that this is a theorem awaiting proof. (This figure is impossible for some combinations of parameters.) Inversion, as in Steiner's, would not work for proof. I don't know the proof of Poncelet's, so I would not know how to proceed here, not that the same method would necessarily work. But if anyone else discovers this, don't get too excited too soon. But it would be astounding if it were true. Comments are very welcome.



