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Topic: New apparent porism - real?
Replies: 2   Last Post: Nov 23, 2003 7:45 PM

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Steve Gray

Posts: 129
Registered: 12/4/04
New apparent porism - real?
Posted: Jul 29, 2002 1:06 AM
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(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.

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