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.

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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Also for questions concerning Soddy's hexlet.

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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 equal. This is problem 6 in this article. I am…
Larry
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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 two other circles. What is the length of each side…
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sangaku - a geometrical puzzle

Find the radius of the circles if the size of the larger square is 1x1. Enjoy! (read about the origin of sangaku)
stevenvh
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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. Rothman). It was given by F. Hidetoshi to the…
Piquito
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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 it just touches the inside of the container…
dsg
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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 $$\frac{1}{\sqrt{R_r}}=\frac{1}{\sqrt{R_b}}+\frac{1}{\sqrt{R_g}}\,.\quad (1)$$ And…
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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 triangle ($∠ABC = 90$), where $AB = 9$ and $BC = 12$.…
Eames
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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 almost solved the problem. I have found the area of…
A.E
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Sangaku: to prove one of the intangents is parallel to $BC$

Given an acute triangle $\triangle ABC$ whose incircle is $I(r)$. Let $O(R)$ be the circle through $B$ and $C$ and which touches $I(r)$ interiorly. Show that the circle $P(p)$ which is tangent to $AB$, $AC$ and $O(R)$ (externaly) is such that one…
hellofriends
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How these circles are congruent?

Here is a problem involving curvilinear incircles and mixtilinear incircles. Let a triangle$\triangle$$ABC$ have circumcircle $\gamma$.It's A-Excircle tangency point at side$BC$ is $D$ Let $\gamma_1$ be the circle tangent to $AD$,$BD$,$\gamma$…
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Exploring a Sangaku problem: proving a dilated circle is circumcircle

$$\Delta ABC \text{ is an equilateral triangle with } D \text{ being the midpoint of } BC \text{. } \Delta DEF \text{ is also an } \\ \text{equilateral triangle such that } E, F \text{ are on minor arc } BC \text{ of the circumcircle of } \Delta…
highgardener
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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& +8\,c^2 ( 3\,c^2+3\,c+2 ) t -4\,c^3…
g.kov
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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: this segment does not have to be a semi-circle).…
user1301930
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Equilateral triangle and very peculiar inscribed tangent circles

The problem is to find the length of the size of the equilateral triangle below I found one equation: Let $R$ be the radius of the big circle whose red arc touches the two purple circles. Let $A$ be the triangle vertex on which the red circle…
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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 Geometry. I've been working on a particular…
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