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I know this page is inadequate, but I have dithered for several months over it, and I finally decided to plunk down the nub of the matter, and hopefully attract some assistance in filling it out with diagrams and references.
Of course, I have details in hand of the claimed proof. I did up a bunch of MS paint pages with the formulas and diagrams, but these are not suitable for a wiki article. Also, note that there is a "cut-the-knot" [1] interactive demo of the theorem which is actually pretty nice, although as far as I can tell, neither that source nor any other comes close to stating the fundamental result that I found.--L mammel (talk) 06:41, 21 May 2008 (UTC)
Here are two links to notes on the proof: page1[2], page2[3]--L mammel (talk) 00:03, 22 May 2008 (UTC)
![[2]](http://www.geocities.com/l_mammel@att.net/sangaku.PNG){kind=link}
![[3]](http://www.geocities.com/l_mammel@att.net/sangaku2.PNG){kind=link}
After some hours of self-tutoring on wiki-editing tips and inkscape usage, I hope I have something approaching a reasonable wikipedia article. Any comments about the content and format of the article are welcome. --L mammel (talk) 17:50, 26 May 2008 (UTC)
I could not find the lemma you refer to. What is it? I also think that your claim as to where the theorem belongs is too sweeping. If you insist it may also belong to analysis, but to exclude geometry is plain thoughtless and, perhaps, arrogant. Alexb@cut-the-knot.com (talk) 03:09, 5 June 2008 (UTC)
I changed the article to state the lemma explicitly. Perhaps my claim that the theorem "belongs to analysis" is too sweeping, since the "equal incircles relation" is derived geometrically. I do claim though that the mapping of tan to sinh, which is the nub of the theorem, is irreducibly analytical, and this mapping lays the theorem bare to the understanding in a way that would be impossible to do with constructive geometry. --L mammel (talk) 04:43, 5 June 2008 (UTC)
Well, I do not know if you can prove your latest claim, but I assure you that even if your lemma may somehow be expressed geometrically, the formulas you have obtained would not lose their value nor shed their analytic nature. So I think that claiming the place of the theorem in analysis is rather superfluous and unnecessary. I am mystified by your insistence. Alexb@cut-the-knot.com (talk) 03:56, 6 June 2008 (UTC)
It's significant to me because my chain of reasoning actually started from the realization that "there must be a continuous scaling function" which determines the equal incircles property. I reasoned that the radii of the defining incircles could be made infinitesimally small, and the equal incircles relation could be integrated to provide this function. I derived the integrand as indicated in my page1 [4] link above, and performing the integral immediately provided the parameterization in terms of sinh[5]. It seems to me that the theorem is completely and uniquely explicated by this simple piece of analysis.--L mammel (talk) 04:47, 6 June 2008 (UTC)