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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.

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.

45 questions
70
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6 answers

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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61
votes
4 answers

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…
vamika2010
  • 721
44
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4 answers

Crazy fact(?) about circles drawn on base of triangle between cevians: they always fit, no matter what their order?

Take any triangle, and draw any number of cevians from the top vertex to the base, with any spacing between the cevians. In each sub-triangle thus formed, inscribe a circle. Now rearrange the order of the circles from left to right (but don't…
Dan
  • 35,053
26
votes
4 answers

Are these circles the same size? (Sangaku problem with a semicircle and two pairs of circles in an equilateral triangle)

The diagram shows an equilateral triangle, a black semicircle, two congruent red circles and two congruent green circles. Wherever things look tangent, they are tangent. Let $g=\text{radius of green circles}$, $r=\text{radius of red…
Dan
  • 35,053
20
votes
5 answers

Show that the radii of three inscribed circles are always in a geometric sequence

A triangle is inscribed in a circle so that three congruent circles can be inscribed in the triangle and two of the segments. Each circle is the largest circle that can be inscribed in its region. Keeping the shortest side (in red) fixed, move the…
Dan
  • 35,053
20
votes
6 answers

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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17
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5 answers

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 $\triangle ADE$, $\triangle DBE$ and $\triangle EBC$ are…
Dan
  • 35,053
17
votes
5 answers

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…
Ataulfo
  • 32,657
16
votes
2 answers

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
  • 1,471
16
votes
3 answers

"The Tiger's Paw" (Sangaku problem with six circles in an equilateral triangle, show that the ratio of radii is three to one.)

In the diagram below, the triangle is equilateral; the four black circles are congruent; wherever things look tangent, they are tangent. Show that the ratio of the green to red radii is three to one. I have found a solution, which I will post…
Dan
  • 35,053
16
votes
1 answer

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 tangent to $\cal E$. Let $\cal C$ be the circle tangent…
timon92
  • 12,291
15
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6 answers

Is there a geometrical diagram in which it is evident that two circles' radii have ratio $1:11$?

There are geometrical diagrams in which it is evident* (to a skilled geometer) that two circles' radii have a certain integer ratio. For example, for the following diagram, it is evident that the ratio of the brown circles' radius to the largest…
Dan
  • 35,053
15
votes
1 answer

Sangaku problem: Show that five circles have equal radii.

Inspired by Sangaku problems such as these, I created the following Sangaku problem. Consider the following diagram. Description: The diagram shows the circle $x^2+y^2=1$, two red chords with equations $y=\pm mx+k$ where $m>1$ and $0
Dan
  • 35,053
15
votes
5 answers

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…
Sebastien B
  • 954
14
votes
3 answers

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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