Sacred Mathematics

Part 1: F.3 Sci and Math Competition

Some Variations

Solution for 2

Solution for 5

Solution for 3

商高定理

青 出

青 入

青出

青入

朱出

朱入

青 朱

Euclid II.14

F.3 Textbook Irrational No.

36

9a

b

r

Part 2: F.4 Circles & Tangents

21

111

RRr

212 RRb

36

9b

c

b = 36 – 9 = 27

c = (452 – 272) = 36

369

1 2

2 21 (36 ) (36 ) 12r r r

2 22 (9 ) (9 ) 6r r r

12 6 36

4

r r

r

Sangaku Problem

9

4

)21(214

1

4

1

)2

1()

2

1(

21)1(

22

222

22

r

rrrrr

xrr

rrrx

1

r

x

Descartes Circle Theorem

1830 Hasimoto Masataka 橋本正孝 / 1643 Descartes's original drawing

Preliminary Result

))(( 3121

2212

rrrr

rAB

Preliminary Result

kr3r2

r1r1θ

21

221

21

3121

2231

221

223

22

2))((2

)()(cos

)(

r

ABrr

rrrr

krrrr

rrk

))(( 3121

2212

rrrr

rAB

Japanese Version of DCT (1)

))((

4

))((

4

))((

4

))((

)2(

332

22

32

331

12

32

3231

212

3

3231

221

232

rtrr

trrBC

rtrr

trrAC

rrrr

rrr

rrrr

rrrAB

Japanese Version of DCT (2)

))(()(

4

4

2

3322

31

232

21

23

222

3

rtrrrr

trrr

r

ACABAH

AB

AH

r

AC

CCAABH

Japanese Version of DCT (3)

))((1

))((

4

3221

21

331

12

3

222

rrrr

rr

rtrr

trr

AHACCH

CHBCBCAC

ACBCABCBCACAB

2

cos222

222

Japanese Version of DCT (4)

133221321

321321323121

321

321321323121

321

1112

1111

)(2

)(2

rrrrrrrrrt

rrrrrrrrrrrr

rrrT

rrrrrrrrrrrr

rrrt