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 O_{1}(r_{1}) and O_{2}(r_{2})
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} = 4r_{1}r_{2}. 

Solution 
Problem 1.1.1
A circle O_{3}(r_{3}) 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
O_{4}(r_{4}) touches l, O_{3}(r_{3})and
O_{1}(r_{1}); a circle O_{5}(r_{5})
touches l, O_{4}(r_{4})and O_{1}(r_{1}),
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 O_{1}(r_{1}), O_{2}(r_{2}),
and O_{3}(r_{3}) touch each other externally.
The line l is tangent to O_{1}(r_{1})
and parallel to the exterior common tangent m to O_{2}(r_{2})
and O_{3}(r_{3}) which does not intersect
O_{1}(r_{1}).
Problem 1.2.4
Two circles O_{1}(r_{1}) and O_{2}(r_{2}),
r_{1}> r_{2}, touch each other externally
and the line l is a common tangent. The line m is parallel
to l and touches the circle O_{1}(r_{1}).
The circle O_{3}(r_{3}) touches m
and the two given circles externally.
Show that r_{1}^{2} = 4r_{2}r_{3}. 

Solution 
Problem 1.2.5
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