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
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.
Problem 1.1.1
A circle O3(r3) is externally tangent
to line l and the two circles described in Problem
1.1.0.
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.
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).
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.
Problem 1.2.5
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