Talk:Japanese theorem for concyclic polygons

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A note on my addition. This theorem came to my attention from the Science News article, and I was intrigued by the invocation of Carnot's theorem. I felt there had to be a simpler derivation, and when I found the animated proof of the quadrilateral theorem linked from the Wikipedia entry on the Cyclic quadrilateral, I was sure this was just a step away. With a few hours of brain strain, I came up with a simple proof, which really involves no construction further than the parallelogram mentioned in my edit, except for extending its sides.

Here is an MS Paint diagram illustrating the proof based on the construction presented in the cited animation.

L mammel (talk) 05:55, 26 March 2008 (UTC)

Well, there may be some disagreement about which proof is simpler. I proved this theorem using fairly simple trigonometric identities, and a respectable if somewhat eccentric mathematician I showed the argument to thought the Carnot argument was simpler. Michael Hardy (talk) 19:08, 29 March 2008 (UTC)

But the proof from the quadrilateral theorem doesn't use anything beyond constructive Euclidean geometry, and well, some inductive reasoning. If we admit Carnot's theorem, then we are allowing the use of an advanced tool. The argument may be simpler in the sense that using a power tool is simpler than using hand tools. It's application is more analytical than geometric in its reliance on the cancellation of signed values. L mammel (talk) 00:27, 30 March 2008 (UTC)

... actually, let me make a bold surmise. I guess we all know that the original Sangaku limited itself to the fan-shaped triangulations where all the diagonals shared one node. I had been thinking that the original reasoning very likely stemmed from the quadrilateral theorem, since the quadrilateral case of the polygon theorem is an easy and natural extension of it. But then there is the matter of the inductive thinking involved in the proof of the general case based on it. This is not so hard, but I think "we moderns" have been extensively trained in that sort of thinking, particularly in our computing education, so it would be questionable whether the original theorem involved a recursive formulation.

However, the limitation to the fan-shaped form makes the application of the quadrilateral case into a simple procedure, essentially identical for a polygon of any number of sides. By starting on one side and applying the quadrilateral rule once in succession to each diagonal, you can "walk" the common node over to the adjacent vertex, and of course the whole procedure can be repeated to show the equivalence of each of the N partitions under the summation of the inradii.

So I would put forward the conjecture that this, or something close to it, was the form of the original thinking behind the Sangaku. L mammel (talk) 04:39, 30 March 2008 (UTC)

At Incenters in Cyclic Quadrilateral: What is this about? we find the following:

This is #3.5 from Fukagawa and Pedoe's collection. It was written on a 1880 tablet in the Yamagata prefecture. H. Fukagawa has noticed that

   rABC + rACD = rABD + rBCD

This implies that the lines through the incenters parallel to the diagonals form a rhombus. You may observe that the arc bisectors of the arcs subtended by the sides of the quadrilateral serve as the diagonals of the rhombus.

... although the inference is reversed, this does connect the quadrilateral theorem with the quadrilateral case of the polygon theorem. BTW, I hadn't observed that fact about the diagonals of the rhombus. L mammel (talk) 08:30, 30 March 2008 (UTC)