1. 74
SAN-GAKU
,
do (1603 – 1867), -
,
.
, .
.
-
-
. Sangaku. Sangaku
-
.
. Sangaku
« ».

2. SAN-GAKU, 75
-
. -
-
: «
».
,
Hidetoshi Fukagawa,
-
-
.
Sangaku
.
-
-
. 1989
Daniel Pedoe
Sangaku
. -
«JAPANESE TEMPLE GE-
OMETRY PROBLEMS».

3. 76 . 4
Sangaku
George Berzsenyi QUANTUM ( / 1995)
« ».
.
( . -
)
.
.
.
, , .
,
, . . .
-
. .

4. SAN-GAKU, 77
. Hidetoshi Fukagawa
: Fukagawa, H. and
Sokolowsky, D. Traditional Japanese Ma-
thematics Problems from the 18th and
19th Centuries, Singapore: Science Culture
Technology Press,
-
.
Hidetoshi Fukagawa
.Hidetosi Fukagawa 61 -
- 2004
Sangaku . -
’
.
« ». ..
3.1
-
.
.
-
.
Ä Ã
ÂÁ Æ
Ç
Ä'
Á'
Å

5. 78 . 4
3.1.1
. -
( , ) ( , ) .
( -
).
( , )
-
.
( .
3 39
,
5 320
)
Ê
Ï
È
Á
Ç
Â
Å ÆÄ Ã
3.1.2
3.1.1.
r ( ,
r)
( . r
16
)
Ê
Ï
È
Á
Ç
Â
Å ÆÄ Ã
3.1.3
-
.
,
.
( .
2a
r
3 5 8
)
Å
ÌÁ Â
Ä Ã

6. SAN-GAKU, 79
3.1.4
, , ,
, , -
-
.
,
.
Ä Ã
BA
Á'
Â'
Ä'
Ã'
( .
( 3 1)
r
4
)
3.1.5
,
.
.
.
. . .
-
Æ
Å
È
Ç
MA B
ÍÄ Ã
3.1.6
.
.
. . . .
.
. . . .
-
-
.
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È
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Ä Ã

7. 80 . 4
3.1.7
-
.
.
( .
1 2 1
2
)
É Ï
A B
ÅÄ Ã
–
[1] QUANTUM / 1995 « » G.Berzsenyi
.46
[2] H. Fukagawa & D. Pedoe "Japanese Temple Geometry Problems Sangaku"
Winnipeg, Canada, 1989
[3] T. Rothman "Japanese Temple Geometry" SCIENTIFIC AMERICAN, May 1998
[4] http://www.wasan.jp/english/
( SANGAKU)
[5] http://www2.gol.com/users/coynerhm/0598rothman.html
( 3. )
[6] http://www2.gol.com/users/coynerhm/0598rothmanbox1a.html
( )
[7] http://www.mfdabbs.pwp.blueyonder.co.uk/Maths_Pages/SketchPad_Files/
Japanese_Temple_Geometry_Problems/Japanese_Temple_Geometry.html
[8] mailto:RXW05750@nifty.com ( . Fukagaua)
,
.

8. 81
,
Erdös-Mordell
,
1
. x1, x2, x3
d1, d2, d3 -
-
: x1 + x2 + x3 2(d1 + d2 + d3)
-
Erdös-Mordell, 10 -
. Erdös (1935),
(1937) Mordell Barrow
Kazarinoff (1945), Bankoff (1958), Oppenheim (1961) .
:
, , ,
, 1 -
-
d1, d2, d3
-
:
2 = 1 = d1 + d2 + d3
x1 + d1 1 -
:
x1 + d1 1 = d1 + d2 + d3
x1 d2 + d3 (1)
d1 .
d2
d1
d3
õ1
x3
x1
x2 Ï
Â'
Ã'
ÃÂ
Á

9. 82 . 4
(1)
:
x1 d2 + d3 1 2 3x d d
' '
.
:
2 3 1
3 1 2
x d d
x d d
:
1 2 3 1 2 3x x x d d d
d1, d2, d3 > 0
2 . . :
x1 + x2 + x3 2(d1 + d2 + d3).
= = .
Erdös-Mordell
d1, d2, d3 , , OAB
, , = 2a, = 2b, = 2c.
2. -
[1].
a, b, c a + b + c = x, y, z
xyz > 0 :
yz zx xy
x a + y b + z c + +
2x 2y 2z
(2)
:
(x z a + y z b xy)2
+ (xz a yz b)2
0, :
y2
z2
+ z2
x2
+ x2
y2
+ 2xyz2
(a + b) 2x2
yz a 2xy2
z b 0
a + b + c = :
2x2
yz a + 2xy2
z b + 2xyz2
c y2
z2
+ z2
x2
+ x2
y2
2xyz :