# Questions tagged [sangaku]

Sangaku are Japanese geometrical problems or theorems on wooden tablets which were placed as offerings at Shinto shrines or Buddhist temples during the Edo period.

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### Six touching circles inside a seventh imply $a+b+c=r$

This is from another question which I started answering but which has been closed before I could finish my answer. Right now there is still a gap in my answer, so it is now my turn to ask about this ...
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### how to solve this Chosekijio sangaku with three equilateral triangles and three circles?

please, how to solve this particularsangaku? Here, sun is a japanese measure unity. sangakus are part of the japanese tradition and are of interest of math problem solvers and enhusiasts and i've ...
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### Self-made Sangaku-style geometry problem involving chords and inscribed circles

In the diagram, circles (or disks, if you like) of the same color have the same radius. (For an explicit description of the diagram, see below.) Let $g=$ radius of the green circles, $r=$ radius of ...
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### What is the size of each side of the square?

The diagram shows 12 small circles of radius 1 and a large circle, inside a square. Each side of the square is a tangent to the large circle and four of the small circles. Each small circle touches ...
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### Chain of circles internally tangent to an ellipse.

I tried to get an answer to this question (which was hastily closed) but couldn't find a proof, so I decided to ask it again, adding some of my efforts. Suppose we have a finite sequence of $n$ ...
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### Japanese Temple Geometry Problem: Two tangent lines and three tangent circles.

I am working on my Senior Thesis for my Bachelor's Degree in Mathematics. My project involves Japanese San Gaku problems, and moving said problems from Euclidean Geometry to Spherical and Hyperbolic ...
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### Solving a sangaku circle problem using a system of equations

From the question "Sangaku Circle Geometry Problem": Given $a$ and $b$, find $c$. (The enclosing circular segment is not necessarily a semicircle.) The answer is I'm curious how one would ...
1 vote
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### Approximating function for the root of quintic polynomial

Related to An ancient Japanese geometry problem As illustrated in the other question, one branch of the solution to the equation \begin{align} 16\,t^5 -8\,c (5\,c +2) t^4 +c^2 (25\,c^2+20\,c + 36)...
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### Is there a way to reduce a specific quintic to cubic?

A polynomial in two variables, $t$ and $c$, is quintic in $t$ and quartic in $c$: \begin{align} 16\,t^5 -8\,c (5\,c +2) t^4 +c^2 (25\,c^2+20\,c + 36) t^3& \\ -4\,c (11\,c^3+8\,c^2+5\,c+2) t^2&...
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### Japanese Temple Problem From 1844

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 ...
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### Japanese Temple Geometry

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 ...
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### Sangaku - Find diameter of congruent circles in a $9$-$12$-$15$ right triangle

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 ...
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### Geometry sangaku puzzle, incribed circle circle/triangle/square

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 ...
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### Sangaku Circle Geometry Problem

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: ...
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### A circle tangent to an ellipse

A friend of mine showed me the following problem: Let $\cal E$ be an ellipse whose semi major axis has length $a$ and semi minor axis has length $b$. Let $\ell_1, \ell_2$ be two parallel lines ...
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### An ancient Japanese geometry problem: Three circles of equal radius inscribed in an isosceles triangle.

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. ...
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### Difficult Recurrence

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 ...
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### Sangaku: Find the Radii of the Inner Circles

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 ...
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### apollonian circles: why are radius and center dual?

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 ...
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### Sangaku. How to draw those three circles with only a ruler and a compass?

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}{\...
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### Sangaku: Show line segment is perpendicular to diameter of container circle

"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 ...
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