Japanese Temple Geometry
Background Figure is Japanese Traditional Pattern "KAGOME"
‚e‚t‚j‚`‚f‚`‚v‚`@‚g‚‰‚„‚…‚”‚‚“‚ˆ‚‰@•@‚g‚n‚q‚h‚a‚d@‚j‚‚š‚•‚Ž‚‚’‚‰



0


SANGAKU PROBLEM 0.
( The Soddyfs Hexlet)


The spheres P and Q lie inside O, touch each other,and also touch O. There is a loop of contact spheres C_i(r_i)(i=1,2,3,c,n) which all touch P, Q, and O.

Show that r_7=r_1 and n=6.



Ani. gif file v.0 (219KB)

Ani. gif file v.1 (221KB)

Ani. gif file v.2 (218KB)

Ani. gif file v.3 (693KB)


Animation gif file v.4 (686KB)



1


SANGAKU PROBLEM 1.

The circles P and Q touch externally and the line at the points A and B. Fix the points A, B, on the line, and vary tow radii of P and Q. Consider the circle R touches P, Q and the line.

Show that the envelope of the circle R describes the circle passing through A and B, and the radius (5/8)AB.


GRAPES file

‚Q‚ƒ‚‰‚’‚ƒ‚Œ‚…Q‚…‚Ž‚–‚…‚Œ‚‚‚…ED‚‡‚‚“




Function Graphing Software
?? GRAPES ??

Animation gif file(210KB)



2


SANGAKU PROBLEM 2.

Given an ellipse, draw an inscribed rhombus.

Show that the rhombus inscribes a fixed circle.




GRAPES file

‚‚–‚‚ŒQ‚’‚ˆ‚‚‚‚‚•‚“‚dD‚‡‚‚“




Function Graphing Software
?? GRAPES ??

Animation gif file(101KB)



Animation gif file Ver.2(82KB)



3


SANGAKU PROBLEM 3.

An ellipse O(a, b) and a circle U touch each other externally, and both a pair of parallel tangent lines.

Show that the distance of two circles; OU=a+b.


GRAPES file

‚‚–‚‚ŒQ‚ƒ‚‰‚’‚ƒ‚Œ‚…‚dD‚‡‚‚“




Function Graphing Software
?? GRAPES ??

Animation gif file(124KB)



4


SANGAKU PROBLEM 4.

Two touching circles C and Q touch a line at A and E. Let BD (B=A) be a diameter of C and let the tangent from D to Q touch it at S.

Show that BD=DS.
This theorem holds when circle C don't touch the line.


GRAPES file

‚Q‚b‚P‚k‚dD‚‡‚‚“




Function Graphing Software
?? GRAPES ??

Animation gif file(109KB)




5


SANGAKU PROBLEM 5.

An ellipse O(a, b) touches a circle O'(r) internally at two points, and a diameter AB of the circle touches the ellipse at an endpoint of its minor axis. The line tangent to the ellipse at the other endpoint of its minor axis meets the circle at P and Q.

Show that AQ=2a.


GRAPES file

‚ƒ‚‰‚’‚ƒ‚Œ‚…Q‚…‚Œ‚Œ‚‰‚‚“‚…‚dD‚‡‚‚“





Function Graphing Software
?? GRAPES ??


Animation gif file(67KB)




6


SANGAKU PROBLEM 6.

The points A, B, C, and D lie on a line, and the circle P has AB as diameter and the circle Q has CD as diameter. Tangent lines are drawn from A to Q and from D to P. Within the first pair and touching them and the circle P internally at B we draw the circle of radius r_1 and within the second pair and touching Q at C internally we draw the circle of radius r_2.

Show that r_1=r_2.


GRAPES file

‚ƒ‚‰‚’_‚”‚‚Ž‚Q‚s‚dD‚‡‚‚“




Function Graphing Software
?? GRAPES ??


Animation gif file(121KB)




7


SANGAKU PROBLEM 7.

30 small balls of radius r cover vone big ball of radius R where each small ball touches four other small balls as shown in the figure.

Show that R=(square root of 5)r.


PDF file
sanpoujojutsu19.pdf
This paper provided in Japanese only.



GRAPES file
‚R‚O‚‚‚‚Œ‚Œ‚“‚dD‚‡‚‚“



Function Graphing Software
?? GRAPES ??

Animation gif file(29KB)



30 balls photo(55KB)










Function Graphing Software
GRAPES 6.75
(GRAph Presentation & Experiment System)
for Windows95,98,Me,NT4,2000,XP

GRAPES (GRAph Presentation & Experiment System) allows you to draw the graphs and the loci of most of the functions which appear in the upper secondary school curriculum, and to analyze them from diverse aspects.



There is five language version.
₯JAPANESE ₯ENGLISH ₯SPANISH ₯PORTUGUESE ₯BOSNIA and Herzegovina




go to top page







go to Japanese Math