San Gaku:
Japanese Temple Geometry Problems

by Kevin Mirus



San Gaku is the term given to a collection of theorems in Euclidian geometry produced by people of all social classes primarily during the Edo period in Japan. These theorems were drawn in color on wooden tablets that hung from the rooves of shrines and temples in local precincts. A collection of some of these problems and their solutions are shown here. The problems are taken from the excellent book by H. Fukagawa and D. Pedoe Japanese Temple Geometry Problems: San Gaku, The Charles Babbage Research Centre, 1989. The drawings and solutions have been worked out by Kevin Mirus.

Table of Contents


 
Chapter I: Circles
Section 1.1: Two Circles
SG010100P00.gif
Problem 1.1.0
SG010101P00.gif
Problem 1.1.1

Problem 1.1.2
SG010103P00.gif
Problem 1.1.3

Problem 1.1.4
 
Section 1.2: Three Circles
 
Problem 1.2.1

Problem 1.2.2
SG010203P00.gif
Problem 1.2.3
SG010204P00.gif
Problem 1.2.4

Problem 1.2.5

 
 
 

Chapter I: Circles

Section 1.1: Two Circles

Problem 1.1.0

The circles O1(r1) and O2(r2) are externally tangent to each other and to the line l at the points A and B as shown in the illustration below.
 
Show that (AB)2 = 4r1r2.
SG010100P01.gif
Solution

Problem 1.1.1

A circle O3(r3) is externally tangent to line l and the two circles described in Problem 1.1.0.
 
Show that SG010101P01.gif.
SG010101P02.gif
Solution

Problem 1.1.2

Problem 1.1.3

Consider the circles in Problem 1.1.1. A circle O4(r4) touches l, O3(r3)and O1(r1); a circle O5(r5) touches l, O4(r4)and O1(r1), and so on.
 
Show that SG010103P01.gif.
SG010103P02.gif
Solution

Problem 1.1.4

Section 1.2: Three Circles

Problem 1.2.1

Problem 1.2.1

Problem 1.2.3

Three circles O1(r1), O2(r2), and O3(r3) touch each other externally. The line l is tangent to O1(r1) and parallel to the exterior common tangent m to O2(r2) and O3(r3) which does not intersect O1(r1).
Show that SG010203P01.gif.
SG010203P02.gif
Solution

Problem 1.2.4

Two circles O1(r1) and O2(r2), r1> r2, touch each other externally and the line l is a common tangent. The line m is parallel to l and touches the circle O1(r1). The circle O3(r3) touches m and the two given circles externally.
Show that r12 = 4r2r3.
SG010204P01.gif
Solution

Problem 1.2.5



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