NOTE: This very difficult problem of elementary geometry has an ancient Japanese source (See “Sacred Mathematics: Japanese Temple Geometry”. Princeton University Press, 2008, by F. Hidetoshi & T. Rothman). It was given by F. Hidetoshi to the Spanish international journal “Revista de la O. I. M....

My attention was brought to a sangaku problem in this book by Ubukata Tou. It shows this figure: The question asks us to find the diameter of the circles (both circles are congruent) in a right triangle ($∠ABC = 90$), where $AB = 9$ and $BC = 12$. It also says that the diameter of the two circl...

Sangaku (算額) are Japanese geometric puzzles written on wooden tablets over 150 years ago. There have been several previous puzzles, but I didn't see this one. Find the radii of the two inner circles in terms of $x$:

Find the radius of the circles if the size of the larger square is 1x1. Enjoy! (read about the origin of sangaku)

I'm having difficulties with this Sangaku problem and was hoping for some help! Five circles (1 of radius c, 2 of radius b, and 2 of radius a) are inscribed in a segment of a larger circle (note: this segment does not have to be a semi-circle). Given a and b, find c. For example, if a = 72 and...

"From a 1803 Sangaku found in Gumma Prefecture. The base of an isosceles triangle sits on a diameter of the large circle. This diameter also bisects the circle on the left, which is inscribed so that it just touches the inside of the container circle and one vertex of the triangle. The top circle...

Hello I am trying to solve a geometry puzzle, its been 30 years since I was in school and I struggled with maths! I would love to get some help to find out what the radius of the bigger circle is if the radius of the smaller circle "乙" is 3.06. Are you clever enough to figure this one out? What...

I found in a book of Sangakus the following problem. Let $R_b$, $R_g$ and $R_r$ the radiuses of the blue, green and red circles $C_b$, $C_g$ and $C_r$. Prove that $$\frac{1}{\sqrt{R_r}}=\frac{1}{\sqrt{R_b}}+\frac{1}{\sqrt{R_g}}\,.\quad (1)$$ And this I can do. But then I would like to draw t...

I am trying to solve a Sangaku problem. The blue circles have radii one. The goal is to find the total area of all the other circles (the three sequences of circles repeat ad infinitum). I have almost solved the problem. I have found the area of the red circle, and the total area of all circl...

This figure suggests the radii and centers (regarded as complex numbers) of the Soddy circles satisfy the same equation: $$ a^2 + b^2 + c^2 + d^2 = \frac{1}{2} (a + b + c + d)^2$$ How can the circle and radius be dual in this particular sangaku problem? http://dl.dropbox.com/u/17949100/soddy.png

Hello, I was trying to solve this problem using descarte circle theorem for my maths report. I looked through the solution but I don't understand the part in the answer, where it says the two solutions are pn+1, pn-1. Can someone explain it for me. Thanks!

I recently learnt a Japanese geometry temple problem. The problem is the following: Five squares are arranged as the image shows. Prove that the area of triangle T and the area of square S are equal. This is problem 6 in this article. I am thinking about law of cosines, but I have not been...

A user created the sangaku tag then edited ten questions to include that tag, all in a short span of time. The user then created descriptions for that tag. While some may see the editing of ten questions in a short span of time as slightly inappropriate, I personally don't see anything wrong wi...

I did this. Sorry if this was inapproriate. I know it's not a good idea to edit many question in a short time span, but I thought 10 was acceptable. With regard to the tag creation, however, I was not aware that it's inappropriate. I'll take note of the various opinions expressed here, for the ...