I've been working on a geometric sangaku problem and found an analytic solution, but this is too complex.
The problem is to determine the squares of maximum and minimum area inscribable in the region determined by two outer tangent circumferences and one of the tangents common to both.
Even determining the square with one of its sides in the common tangent and the other two vertices one in each circumference, is no longer a trivial task (except in the case where the circumferences are of equal radius).
I have worked on direct algebraic approaches and I have also tried it by means of inversion transformations, but, despite the simple appearance of the problem, it does not seem to have a simple solution
Can anyone suggest me any other method or strategy to address this problem?
Thank you very much.
It seems to be a very nice problem. But I am sorry that I can not
suggest any method right now.
Please allow me to ask the followings.
>Even determining the square with one of its sides in the common tangent and the other two vertices one in each circumference, is no longer a trivial task (except in the case where the circumferences are of equal radius).
The trivial case can be found in a sangaku, but I have never seen the
general case as in the first figure. Is this your generalization?
Thank you very much.
Thanks Hiroshi;for your answer.
No, it was proposed in a sangaku as it is described by me.. I have solved it, but the solution takes many pages. I have explored different approaches, but they all end up in very complicated equations. Among them, I have found one that provides algebraic solution, but so complex that in practice it could only be solved numerically. The conclusion I came to after almost exhaustively exploring the problem is that there is no simple solution to it.
When I finish writing the document I will publish it here in RG.
Certainly, it is a provocative problem, difficult to remove from the head.
Have you tried to use the Langrangian?
I am not sure how to formulate the actual Langrangian Function,
but it might help you to figure out a way of solving the problem
--
Yours,
Daniel
Thank you for your suggestion Daniel. Of course I know Lagrange's method, but in order to apply it it is necessary to define the square area function that satisfies the geometric conditions of the problem and that is precisely the greatest difficulty. Perhaps we could try to define the area of the square partially linked to the region in which it must be inscribed, in order to reduce the complexity of the square area function, and impose the remaining conditions as constraints by the Lagrange method. I write this which is the first thing that comes to mind, without having tested at all if this is possible. In any case, judging by the results I have obtained, I believe that there are no shortcuts, that is, there can be no possibility of finding the solution without a great effort.
Thanks Jesús Álvarez Lobo for your comments.
I am looking for the sangaku problem with two circles of different radii.
But I can not.
Hiroshi, I hope to send you my work as soon as possible. Thanks for the interest shown in the problem. Certainly, this problem constitutes a whole field of research, which can encourage the development of more powerful tools to be able to tackle it more efficiently or the search for ingenious strategies.
I am looking forward to seeing your works, Jesús Álvarez Lobo.
Thanks!!
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