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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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9 votes
8 answers
454 views

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 ...
MvG's user avatar
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-2 votes
1 answer
137 views

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 ...
Humberto José Bortolossi's user avatar
9 votes
2 answers
487 views

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 ...
Dan's user avatar
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13 votes
3 answers
595 views

I created a Sangaku-style geometry problem involving an equilateral triangle and three circles. Can you solve it without a computer?

Inspired by this difficult Sangaku problem, I created the following Sangaku-style problem of my own. In equilateral $\triangle ABC$, $D$ is on $AB$, $E$ is on $AC$, and the incircles of $\triangle ...
Dan's user avatar
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8 votes
1 answer
509 views

Japanese Temple Geometry Problem: Radii of inner circles inside quarter arcs

I was able to get the equation for the radius of larger circle but couldn't think for the smaller one. Source: wu riddles
Navdeep Singh's user avatar
1 vote
2 answers
207 views

An equilateral triangle and circle inscribed in a semicircle of radius $1$, find the radius of the circle.

This is a classical Sangaku problem, also known as old Japanese geometry problems, that I found out just recently. The figure shows a semicircle with a smaller circle and an equilateral triangle ...
冥王 Hades's user avatar
  • 3,086
7 votes
3 answers
624 views

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 ...
hellofriends's user avatar
  • 1,740
4 votes
1 answer
226 views

Unique Tangency of circles from Sangaku

A Geometric Sangaku shape here on this wooden board up right from Japan's Buddhist Temples in 1859 during Edo period drew my attention recently, however I've found the exact values of the radius of ...
MasM's user avatar
  • 641
9 votes
1 answer
369 views

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 $\gamma_1$ ...
Saad Junior's user avatar
3 votes
3 answers
189 views

$10$ circles ($2$ large of radius $R$, $6$ small of radius $r$ and 2 small of radius $t$) are enclosed in a square. How we find $r$ in terms of $t$?

Let us embed $2$ large intersecting circles of radius $R$ into a square as depicted by the figure below. These two circles are highlighted green. Into these $2$ circles we embedd $6$ smaller ones of ...
user avatar
2 votes
1 answer
224 views

Prove that the sum of the radii of the circles

$ABCD$ is a cyclic quadrilateral. Prove that the sum of the radii of the circles drawn inside the triangles $\Delta ABC$ and $\Delta CDA$ is equal to the sum of the radii of the circles drawn inside ...
studentsmath's user avatar
11 votes
3 answers
409 views

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 $AB$,...
hellofriends's user avatar
  • 1,740
7 votes
2 answers
326 views

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 ...
highgardener's user avatar
61 votes
4 answers
6k views

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 ...
vamika2010's user avatar
11 votes
4 answers
920 views

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 $n$ ...
Intelligenti pauca's user avatar
6 votes
2 answers
385 views

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 ...
kennethmoore's user avatar
2 votes
0 answers
222 views

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 ...
A. Mueller's user avatar
1 vote
0 answers
303 views

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)...
g.kov's user avatar
  • 13.6k
7 votes
1 answer
248 views

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&...
g.kov's user avatar
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70 votes
6 answers
8k views

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 ...
Larry's user avatar
  • 5,090
4 votes
1 answer
375 views

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 ...
Rosa Kang's user avatar
14 votes
3 answers
2k views

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 ...
Eames's user avatar
  • 251
5 votes
3 answers
1k views

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 ...
PaddySmith's user avatar
7 votes
1 answer
2k views

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: ...
user1301930's user avatar
17 votes
1 answer
2k views

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 ...
timon92's user avatar
  • 11.3k
17 votes
5 answers
2k views

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. ...
Piquito's user avatar
  • 29.9k
11 votes
3 answers
635 views

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 ...
A.E's user avatar
  • 2,473
3 votes
1 answer
1k views

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 ...
jmac's user avatar
  • 191
2 votes
1 answer
378 views

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 ...
cactus314's user avatar
  • 24.5k
15 votes
5 answers
2k views

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}{\...
Sebastien B's user avatar
15 votes
2 answers
2k views

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 ...
dsg's user avatar
  • 1,441
20 votes
6 answers
9k views

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's user avatar
  • 2,686