abcd bc cd - wasan · 2016. 8. 21. · !!!!a 4 e 2 + 2 ah (a 2! q 2)ef + a 2 b 2 f 2! a 2 c2 p 2 q...

16
!"#"! !"##! Formula!" $% &’’()*& (%*+,(-&* . ,&+/.%0’& ABCD 1 . +(,+’& +(,+23*+,(-&* /4& &’’()* .%5 /62+4&* BC .%5 CD1 .* *467%8 9&/ p, q .%5 d -& /4& *&3(:3.;6,1 3(%6, .<&* 6= /4& &’’()*& .%5 /4& 5(.3&/&, 6= /4& +(,+’&1 AB = a > BC = b 1 9&/ D 2 = a 2 b 2 ! p 2 q 2 > G 2 = ab ! D > D + ab ! 2ad ! 2 bd ! 2Gd + d 2 = 0 ?Proof@ A4& =(02,& B (* /4& =(02,& 74&,& /4& =(02,& C 7.* *46,/&%&5 q p (% /4& 5(,&+/(6% 6= /4& *&3(:3.;6, .<(*8 D% /4& =(02& C>’&/ NT = c> AN = e> AT = f > EG = g> EF = h 1 #E HCJ F#E HJO GE CJE FE CDH ! be + af " 2ef = 2 cIO HHI J6,&6K&,1 E H 2 J 2 O 2 G q p E HJO ! 2 cqIO p = q H 2 J 2 HHL M2-*/(/2/& I (%/6 L1 be + af ! 2ef = p H 2 J 2 N.(*& /4& -6/4 *(5&* /6 /4& *&+6%5 )67&,1 p 2 H 2 J 2 2 = b 2 e 2 + a 2 f 2 + 4 e 2 f 2 + 2 abef ! 4be 2 f ! 4 aef 2 HHO P Q,63 /4& =(02,& C1 PQ = fh g > NP = ae g > AG = b 2 + h 2 > AK = a AG g = ab 2 g + ah 2 g > GT = AG ! f > MT = a GT g = a AG g ! af g = ab 2 g + ah 2g ! af g > AP = eh g > KP = AK ! AP = ab 2g + ah 2 g ! eh g > D 2 H 2 = MT ! KP = ! af g + eh g > D 2 J 2 = q PQ + NP ( ) p = fhq gp + aeq gp D% ED 2 H 2 J 2 > H 2 J 2 2 =D 2 H 2 2 +D 2 J 2 2 M2-*/(/2/& &.+41 .%5 .,,.%0&1 H 2 J 2 2 = ! af g + eh g " # $ % & 2 + fhq gp + aeq gp " # $ % & 2 = a 2 f 2 g 2 ! 2 aefh g 2 + e 2 h 2 g 2 + f 2 h 2 q 2 g 2 p 2 + 2 aefhq 2 g 2 p 2 + a 2 e 2 q 2 g 2 p 2 p 2 H 2 J 2 2 = a 2 f 2 p 2 g 2 ! 2 aefhp 2 g 2 + e 2 h 2 p 2 g 2 + f 2 h 2 q 2 g 2 + 2 aefhq 2 g 2 + a 2 e 2 q 2 g 2 HR Q,63 O1 R> b 2 e 2 g 2 + a 2 f 2 g 2 + 4e 2 f 2 g 2 + 2abefg 2 ! 4be 2 fg 2 ! 4aef 2 g 2 ! a 2 f 2 p 2 + 2 aefhp 2 ! e 2 h 2 p 2 ! f 2 h 2 q 2 ! 2aefhq 2 ! a 2 e 2 q 2 = 0 HHS

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Page 1: ABCD BC CD - WASAN · 2016. 8. 21. · !!!!a 4 E 2 + 2 ah (a 2! q 2)EF + a 2 b 2 F 2! a 2 c2 p 2 q 2 = 0 ! &'()!*+ !!!h 2 A = a 2 B = a 2 (p 2! a 2)+!!!!! a 4 E 2 + 2a 2 DEF + a 2

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !"#"!! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !"##!

Formula!"! $%!&''()*&!(%*+,(-&*!.!,&+/.%0'&! ABCD 1!.!+(,+'&! !

+(,+23*+,(-&*!/4&!&''()*!.%5!/62+4&*! !

BC! .%5! CD1!.*!*467%8!

! ! ! ! 9&/! p, q ! .%5! d ! -&!/4&!*&3(:3.;6,1! !

3(%6,!.<&*!6=!/4&!&''()*&!.%5! !

/4&!5(.3&/&,!6=!/4&!+(,+'&1 AB= a>BC= b 1

! 9&/! D2

= a2

b2

! p2

q2

>G2

= ab ! D>!

! ! ! ! ! D + ab ! 2ad ! 2bd ! 2Gd + d2

= 0 !

?Proof@! ! A4&!=(02,&!B!(*!/4&!=(02,&!74&,&!/4&!=(02,&!C!7.*!

*46,/&%&5! !

! !q

p! (%!/4&!5(,&+/(6%!6=!/4&!*&3(:3.;6,!.<(*8! ! !

!

!

!

!

! D%!/4&!=(02&!C>'&/ NT = c> AN = e> AT= f> EG = g> EF = h1!

! ! ! ! ! ! ! ! #EHCJ F#EHJO GE CJEFECDH !

! ! ! ! ! ! ! be + af "2ef =2 cIO ! ! HHI!

J6,&6K&,1! ! ! EH2J2O

2G

q

pEHJO !

! ! ! ! ! ! !2cqIO

p= qH 2J 2 ! ! HHL!

! M2-*/(/2/&!I! (%/6!L1! ! ! ! be + af !2ef = pH 2J 2 !

N.(*&!/4&!-6/4!*(5&*!/6!/4&!*&+6%5!)67&,1!

! ! p2H 2J 2

2

=b2e2+ a

2f2+ 4 e

2f2+ 2 abef !4be

2f !4 aef

2! ! HHO!

P! Q,63!/4&!=(02,&!C1!

! PQ=fh

g> NP =

ae

g> AG =

b

2+h

2> AK =

aAG

g=ab

2 g+ah

2 g>!

! GT = AG ! f>MT =aGT

g=a AG

g!af

g=ab

2 g+ah

2g!af

g> AP =

eh

g>!

! KP = AK ! AP=ab

2g+ah

2 g!eh

g> D2H 2 =MT !KP = !

af

g+eh

g>!

! D2J 2 =q PQ + NP( )

p=fhq

gp+aeq

gp!

D%!ED2H

2J2>! ! H 2J 2

2

= D2H 2

2

+ D2J 2

2

!

M2-*/(/2/&!&.+41!.%5!.,,.%0&1!

! ! H 2J 2

2

= !af

g+eh

g

"

# $ %

& ' 2

+fhq

gp+aeq

gp

"

# $ %

& ' 2

!

! ! ! ! ! ! ! ! =a 2 f 2

g2!2 aefh

g2+e2h2

g2+f 2h2q2

g2 p 2+2aefhq 2

g2 p 2+a 2e2q2

g2 p 2!

p2H 2J 2

2

=a 2 f 2 p 2

g2!2 aefhp 2

g2+e2h2 p 2

g2+f 2h2q2

g2+2 aefhq 2

g2+a 2e2q2

g2HR!

Q,63!O1!R>!

! b2

e2

g2

+ a2

f2

g2

+ 4e2

f2

g2

+ 2abefg2

! 4be2

fg2

! 4aef2

g2

!

! !a2

f2

p2

+ 2aefhp2

! e2

h2

p2

! f2

h2

q2

! 2aefhq2

! a2

e2

q2

= 0 ! HHS!

Page 2: ABCD BC CD - WASAN · 2016. 8. 21. · !!!!a 4 E 2 + 2 ah (a 2! q 2)EF + a 2 b 2 F 2! a 2 c2 p 2 q 2 = 0 ! &'()!*+ !!!h 2 A = a 2 B = a 2 (p 2! a 2)+!!!!! a 4 E 2 + 2a 2 DEF + a 2

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !"#$!! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !"#%!

&! '()*!+,-!./01(-!23!

! ! ! ! ! ! FW=ah

g4 GW =

a2

g4 G 2 W=

qGW

p4!

! ! A2B2

2

= FW2

+ G 2 W2

=a 2h2

g2+q2GW

2

p 2=a 2h2

g2+a 4q2

g2 p 2!

5(,)*617!/8!./092:2=M2N 2

2!! A2B2

2

=q2 a 2h2

g2+a 4q2

g2 p 2

"

# $

%

& ' !

;8!+,-!)+,-(!,<8=3! ! ! (->+<80?-! ABCD = ab 4!

! ! ! ! ! ! ! ! !5(,)*617!/8!./092:2

=a2b 2q 2

p2!

! ! ! ! ! ! ! h2

p2

+ a2

q2

" b2

g2

= 0 ! ! @@A!

B! C8!<==/+/)83!.()*!+,-!./01(-!23!

! ! ! ! ! ! ! ! ! ! E2O=h2

g4 E2 W=

qE2O

p4!

! ! ! E2F2

= E2 W2

+ FW2

=q2E2O

2

p 2+a 2h2

g2=h4q2

g2 p 2+a 2h2

g2!

C8!+,-!./01(-!23!+,-!<(-<!).!(,)*617!D/+,!+,-!7/=-! E2F 3!

! ! ! q2

E2F2

=a2 q2 h2

p 2! ! ! h

2

q2

+ a2

p2

" a2

g2

= 0 ! ! @@E!

F! AGE! ! h2

p2

+ a2

p2( ) + h

2

q2

+ a2

q2( ) ! b2 g2 ! a 2

g2

= 0 !

! ! ! ! ! ! ! ! ! ! p2

+ q2

" b2

" a2

= 0 ! ! @@H!

! IGA!e2

+E! f2

!

! ! ! ! 2efg2

+ abg2

! 2beg2

! 2afg2

+ ahp2

! ahq2

= 0 ! ! @@J!

! EKg2

= a2

+ h2

LMNOPQ4! ! h2

q2

+ a2

p2

! a4

! a2

h2

= 0 !

R-+! A = a2

! q2

4 B = p2

! a2

4! a2

B ! h2

A = 0 ! ! @@S!

! ! ! a2

B ! h2

A + a2

A ! a2

A = 0 ! B + A( )a2 ! a2

+ h2( )A = 0 !

! ! ! ! ! ! ! ! ! a2

C = g2

A ! ! @@@T!

! J! A ! ! 2efg2

A + abg2

A ! 2beg2

A ! 2afg2

A + ahA p2

! q2( ) = 0 !

! U-++/80! ! C = p2

! q2

3!<8=!7167+/+1+-!T4!

! ! ! ! ! 2ef + ab ! 2be ! 2af + AB = 0 !

R-+! AB = a2

! q2( ) p2 ! a2( ) = a2 b2 ! p 2

q2

= D2

4!

! ! ! ! ! 2ef + ab ! 2be ! 2af + D = 0 ! ! ! @@@V!

F! C8!<==/+/)83!.()*!+,-!./01(-!23!

! ! TV= f !d

24 ST=

TV!D2H 2

c= "

af

cgTV+

eh

cgTV4!

! LR = I2K 2 = MT !ST=ab

2g+ah

2 g!af

g+af

cgTV!

eh

cgTV ! ! @@W!

! RS=

TV TU + PN( )c

=fh

cgTV+

ae

cgTV4!

! GM =hGT

g=h

gAG ! f( ) =

bh

2 g+h2

2 g!fh

g4 LO = PQ+ GM !

g

2! !

! K 2O2 =q

pRS+ GM !

g

2

"

# $ %

& ' !

! ! ! ! =fhq

cgpTV+

aeq

cgpTV+

bhq

2 gp+h2q

2gp!fhq

gp!gq

2 p! ! @@X!

Y! C8!Z I2K2O24! ! K 2O2 + I2K 2 !

q2

4= 0 !

U-++/80E = !2cf + 2 f VT+ bc4F = !2eVT+ ac ! <8=!7167+/+1+-!W4X! !

<8=!<((<80-3!

! ! a2

p2

E2

+ 2ahp2

EF + h2

p2

F2

+ h2

q2

E2

!

! ! ! ! ! ! ! ! ! ! ! ! ! !2ahq2

EF + a2

q2

F2

! c2

g2

p2

q2

= 0 !

! a2

p2

+ h2

q2( )E 2

+ 2ah p2

! q2( )EF + h

2

p2

+ a2

q2( )F 2

! c2

g2

p2

q2

= 0 !

U167+/+1+-!A4E4!

! ! ! a2

g2

E2

+ 2ah p2

! q2( )EF + b

2

g2

F2

! c2

g2

p2

q2

= 0 !

[,-8!D-!*1?+/\?]! A = a2

! q2

! +)!6)+,!7/=-7!<8=!7167+/+1+-!T3!

! ! ! ! ! a4

p2

! q2( )E 2

+ 2ah a2

! q2( ) p2 ! q2( )EF !

! ! ! ! ! ! ! ! ! ! ! +a2

b2

p2

! q2( )F2

! a2

c2

p2

q2

p2

! q2( ) = 0

Page 3: ABCD BC CD - WASAN · 2016. 8. 21. · !!!!a 4 E 2 + 2 ah (a 2! q 2)EF + a 2 b 2 F 2! a 2 c2 p 2 q 2 = 0 ! &'()!*+ !!!h 2 A = a 2 B = a 2 (p 2! a 2)+!!!!! a 4 E 2 + 2a 2 DEF + a 2

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !"#$!! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !"#%!

! ! ! ! a

4

E2

+ 2ah a2

! q2( )EF + a

2

b2

F2

! a2

c2

p2

q2

= 0 !

&'()!*+! ! ! h2

A = a2

B = a2

p2

! a2( )+!

! ! ! ! a4E2+ 2a

2DEF + a

2b2F2" a

2c2p2q2= 0 ! ! ,,-!

.! /0!123!(123'!24056!

! ! E = !2cf + 2 f VT+ bc = bc !2 f c! VT( )= bc !2 f e!d

2

"

# $ %

& ' ! ! !

! ! ! = bc ! 2 fe ! df( ) = bc ! 2 "d

2c + e + f( ) ! df{ } = bc ! d c + e( ) ! ,7!

! ! F = !2eVT+ ac = ac !2 eVT = ac !2 e f !d

2

"

# $ %

& ' =ac ! 2 ef !ed( ) !

! ! ! ! ! ! = ac ! 2 "d

2c + e + f( ) ! ed{ } = ac ! d c + f( ) ! ! ,,8!

9:;<1=1:13!7+8! =01(!-6!<311=0>! G = ab ! D ! 405!4''40>36!

! ! a2

b2

c2

! 2a2

bcd c + e( ) + a2d 2

c + e( )2

+ 2abc2

D ! 2acdD c + e( ) !

! ! ! ! ! !2bcdD c + f( ) + 2d 2

D c + e( ) c + f( ) + a2b 2

c2

!

! ! ! ! ! ! ! ! ! ! ! !2ab2

cd c + f( ) + b2d 2

c + f( )2

! c2

p2

q2

= 0 !

?23'3@('36!

! ! cD + abc ! ad c + e( ) ! bd c + f( ){ }2

! d2

G2

c + e + f( ) 2 = 0 !

! ! ! cD + abc " ad c + e( ) " bd c + f( ) " dG c + e + f( ) = 0 ! ! ,,A!

B!d + A!2 !

! D c + e + f( ) + ab c + e + f( ) ! 2ad c + e + f( ) !

! ! ! ! ! ! ! !2bd c + e + f( ) ! 2dG c + e + f( ) + d2 c + e + f( ) = 0 !

! ! ! ! ! ! D + ab " 2ad " 2 bd " 2dG + d2

= 0 !

! ! ! ! ! ! ! ! ! ! ! CDE+ D2

= a2

b2

! p2

q2

+G2

= ab ! D !

! Formula""! F!G='GH3! O1! =0<G'=;3<!123!3HH=I<3!41!1J(!I(=01<!

! ! 405!123!G='GH3! O2! G='G:)<G'=;3<! !

123!G='GH3! O1! 41!123!<4)3!I(=01K!

! ! ! L31! p, q, d1! 405! d

2! ;3! !

123!<3)=M)4N('6!)=0('!4O=<6! !

123!5=4)313'<!(@!123!G='GH3! O1! !

405! O26 O

2H = a+OH = b K!

! ! L31! A = p2

+ q2

+ d2

2

! 4a2

! 4b2

!

! ! ! ! B = !p2

q2

! d2

2

p2

+ q2( ) + 4a2 p2 + 4b2q2+!

! ! P! Ad1q2+ 2d

2q4! d

1

3p2= 0 !

! ! Q! !Bd1

2! d

2

2q4+ 2d

1

3d2p2= 0 !

! ! R! Ad2q2! 2Bd

1+ 3d

1

2d2p2= 0 !

! ! S! 3d2

2q4+ 2Ad

1d2q2! Bd

1

2= 0 !

TI'((@U! P! VW!@('):H4!XY+123!G(M('5=0413<(@!I(=01!(@!140>30GW! !

! ! P x1, y1( )+! ! !

! ! ! ! x1= !

p2

p2

! q2( ) q 2

! d1

2( )2q p2 ! q2( )

!

! ! ! ! y1=1

2

d1

2 p2 ! q4

p 2! q 2

!

?23!G(('5=0413<!(@!123!I(=01! O1Z!

! ! ! ! O1 !p2! q

2( ) q 2! d1

2( )2q

, 0

"

#

$ $

%

&

' ' !

[230!J3!4<<:)3!12=<!1(!;3! O1 n, 0( ) 6! !

\30G3!]O1PH

1^]PO

2M+!

Page 4: ABCD BC CD - WASAN · 2016. 8. 21. · !!!!a 4 E 2 + 2 ah (a 2! q 2)EF + a 2 b 2 F 2! a 2 c2 p 2 q 2 = 0 ! &'()!*+ !!!h 2 A = a 2 B = a 2 (p 2! a 2)+!!!!! a 4 E 2 + 2a 2 DEF + a 2

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !"#$!! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !"#%!

! ! ! !d1

2: n !x1( ) =

d2

2:H1H ! ! H1H =

d2

d1

n "x1( ) ! !

! ! ! !d1

2: y1 =

d2

2:MO2 ! ! ! ! MO2 =

d2y1

d1!

&'()(*+)(,!-'(!.+/+)0123-(4+*!+*!-'(!5+12-! O2! 6!

! ! ! ! ! ! ! O 2 x1 !d2

d1n ! x1( ), y1 +

d2y1

d1

" #

$ % !

! y /.++)0123-(!+*!-'(!5+12-! O2! 6!

! ! ! ! ! ! ! ! a =d1+ d

2

d1

!1

2

d1

2 p2 " q4

p2 " q 2!

! ! ! ! 4 d1

2

a2

p2

! q2( ) ! d1

4

p2

! 2d1

3

d2p2

!

! ! ! ! ! ! ! ! ! ! d1

2

d2

2

p2

+ d1

2

q4

+ 2d1d2q4

+ d2

2

q4

= 0 ! ! 778!

! x /.++)0123-(!+*!-'(!5+12-! O2! 6!

! ! ! b =p2

d1+ d

2( ) p

2

! q2( ) q 2

! d1

2( )2d

1q p2 ! q 2( )

!d2

p2

! q2( ) q 2

! d1

2( )2 d

1q

!

! ! ! ! ! ! ! 4b2

d1

2

q2

p2

! q2( ) = q

2

! d1

2( ) d1 p2

+ d2q2( )

2

!

! ! ! 4b2

d1

2

q2

p2

! q2( ) ! d2

2

q6

! 2d1d2p2

q4

!

! ! ! ! ! ! ! ! d1

2

p4

q2

+ d1

2

d2

2

q4

+ 2d1

3

d2p2

q2

+ d1

4

p4

= 0 ! ! 779!

8!q2

:9! ! 4 d1a2

q2

+ 4b2

d1q2

+ d1

3

p2

!

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! d1d2

2

q2

! 2d2q4

! d1p2

q2

! d1q4

= 0 !

! ! ! ! ! ! ! ! ! ! Ad1q2

+ 2d2q4

" d1

3

p2

= 0 !

;! 8! p2

:9! ! !d1

2

p2

q2

! d1

2

d2

2

p2

+ q2( ) + 4d1

2

a2

p2

!

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! + 4b2

d1

2

q2

+ d2

2

q4

! 2d1

3

d2p2

= 0 !

! ! ! ! ! ! ! ! ! ! ! " d1

2

B " d2

2

q4

+ 2d1

3

d2p2

= 0 !

<! 1( ) !d2

d1

+ 2( ) !2

d1

! ! ! ! Ad2q2

! 2Bd1+ 3d

1

2

d2p2

= 0 !

=! 1( ) ! 2d2+ 2( ) ! ! ! ! ! ! 3d

2

2

q4

+ 2Ad1d2q2

! Bd1

2

= 0 !

Formula!""! >!.1).?(! O1! 124.)1@(4! !

-'(!(??154(!3-!-A+!5+12-4!320!3!.1).?(! !

O2! 124.)1@(4!-'(!(??154(!3-!-'(!43B(! !

! 5+12-C!

! ! ! D(-! p, q, d1! 320! d

2! @(!-'(!4(B1/B3E+),!B12+)!3F(4!320!-'(! !

! ! 013B(-()4!+*!-'(!.1).?(4! O1! 320! O

2, O

2H = aGOH = b C!

! ! D(-! A = p2

+ q2

+ d2

2

! 4a2

! 4b2

G!

! ! ! ! ! ! ! B = p2

q2

+ d2

2

p2

+ q2( ) ! 4 p2a2 ! 4q2b2G!

H! Ad1q2! 2d

2q4! d

1

3p2= 0 !

;! Bd1

2! d

2

2q4! 2d

1

3d2p2= 0 !

<! !Ad2q2+ 2Bd

1! 3d

1

2d2p2= 0 !

=! 3d1

2q4! 2Ad

1d2q2+ Bd

1

2= 0 !

IProofJ! H! KL!*+)BM?3!NO,!-'(!.+/+)0123-(4!+*!-'(!5+12-!+*! ! !

! -32P(2.L! P x1, y1( ) 6!

! ! ! ! ! x1=p2

p2

! q2( ) q2 ! d1

2( )2q p2 ! q 2( )

!

! ! ! ! ! y1=1

2

d1

2 p2 ! q4

p 2! q 2

!

&'(!.+/+)0123-(4+*!-'(!5+12-! O16! O1

p2! q

2( ) q2 ! d12( )

2q, 0

"

#

$ $

%

&

' ' !

Q(--12P! O1 n, 0( )G! 412.(!RO1PH

1SRPO

2MG!

! !d1

2: x1 !n( ) =

d2

2:H1H 2 ! ! H1H 2 =

d2

d1

x1 "n( ) ! !

! !d1

2: y1 =

d2

2: PM ! ! ! ! ! ! PM =

d2y1

d1!

Page 5: ABCD BC CD - WASAN · 2016. 8. 21. · !!!!a 4 E 2 + 2 ah (a 2! q 2)EF + a 2 b 2 F 2! a 2 c2 p 2 q 2 = 0 ! &'()!*+ !!!h 2 A = a 2 B = a 2 (p 2! a 2)+!!!!! a 4 E 2 + 2a 2 DEF + a 2

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !"#$!! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !"%&!

'()*)+,*)-!.()!/,0,*1234.)5!,+!.()!6,23.! O

2! 7!

! ! ! ! O 2 x1 !d2

d1x1 ! n( ), y1 !

d2y1

d1

" #

$ % !

! y ! 0/,0,*1234.)!,+!.()!6,23.! O2! 7!

! ! ! ! ! a =d1! d

2( ) p

2

! q2( ) d1

2

p2

! q4( )

2d1p2 ! q2( )

!

! ! ! ! 4a2

d1

2

p2

" q2( ) " d1

4

p2

+ 2d1

3

d2p2

!

! ! ! ! ! ! ! ! !d1

2

d2

2

p2

+ d1

2

q4

! 2d1d2q4

+ d2

2

q4

= 0 ! ! 889!

! x 0/,,*1234.)!,+!.()!6,23.! O2! 7!

! ! ! b =p2

d1! d

2( ) p

2

! q2( ) q 2

! d1

2( )2d

1q p2 ! q 2( )

+d2

p2

! q2( ) q 2

! d1

2( )2 d

1q

!

! ! ! ! 4b2

d1

2

q2

p2

" q2( ) " d1

2

p4

q2

+ 2 d1d2p2

q4

!

! ! ! ! ! ! ! ! ! !d2

2

q6

+ d1

4

p4

! 2d1

3

d2p2

q2

+ d1

2

d2

2

q4

! 0 ! ! 88:!

! 9!q2

;:! ! 4 d1q2

a2

+ 4d1q2

b2

! d1p2

q2

! d1q4

!

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! d1d2

2

q2

+ 2 d2q4

+ d1

3

p2

! 0 !

! ! ! ! ! ! ! ! ! ! ! Ad1q2

" 2d2q4

" d1

3

p2

= 0 !

<! 9! p2

;:! ! d1

2

p2

q2

+ d1

2

d2

2

p2

+ q2( ) ! 4d 1

2

p2

a2

!

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 4d

1

2

q2

b2

! d2

2

q4

! 2 d1

3

d2p2

= 0 !

! ! ! ! ! ! ! ! ! ! Bd1

2

" d2

2

q4

" 2d1

3

d2p2

= 0 !

=! 2( ) !2

d1

" 1( ) !d2

d1

!

! ! ! Bd1

2

! d2

2

q4

! 2d1

3

d2p2( ) "

2

d1

! Ad1q2

! 2 d2q4

! d1

3

p2( ) "

d2

d1

= 0 !

! ! ! ! ! ! ! " Ad2q2

+ 2Bd1" 3d

1

2

d2p2

= 0 !

>! 2( ) ! 1( ) " 2d2!

! ! ! ! Bd1

2

! d2

2

q4

! 2d1

3

d2p2( ) ! Ad

1q2

! 2d2q4

! d1

3

p2( ) " 2d2 = 0 !

! ! ! ! ! ! ! ! ! ! 3d2

2

q4

" 2Ad1d2q2

+ Bd1

2

= 0 !

!

Formula!"!! '?,!/,3@*A)3.!5BA4*)! !

.,A/(!.()!)CC265)!4.!+,A*!6,23.5!

A,C, E! 431! GD!

! ! '()!124@,34C5! BD! 431! DF! 25! !

,3!.()!5)E20E4F,*!4G25!,+!.()!)CC265)D!

! ! H).! p ! 431! q! I)!.()!5)E20E4F,*!431!E23,*!4G)5!,+! !

! ! .()!)CC265)- AB= aJDE = bJ!

! ! a2

p2

+ q2( )2

+ b2

p2

+ q2( )2

!

! ! ! ! ! ! ! ! ! !2ab p2

+ q2( ) p2 ! q2( ) ! 2p2q4 = 0 !

KProofL! '()!)BA4.2,3!,+!.()!)CC265)J4 x

2

p2+4y

2

q2= 1! ! 889!

! H)..23@! D m, 0( )J! ! A m !a

2,a

2

"

# $ %

& ' J! ! G m +

b

2,b

2

!

" # $

% & !

! M)/4A5)!.()!6,23.! A ! 25!4!6,23.!,3!.()!)CC265)-!5AI5.2.A.)!23.,!9D!

! ! ! ! 4q2

m2

! 4 2 aq2

m + 2a2

p2

+ 2 a2

q2

! p2

q2

= 0 ! ! 88:!

N2E2C4*CO-!5AI5.2.A.)!.()!/,0,*1234.)5!,+!6,23.! G ! 23.,!9-!

! ! ! 4q2

m2

+ 4 2bq2

m + 2b2

q2

+ 2b2

p2

! p2

q2

= 0 ! ! 88P!

:!P! ! !4 2 aq2

m ! 4 2 bq2

m + 2a2

p2

+ 2a2

q2

! 2b2

q2

! 2b2

p2

= 0 !

! ! ! ! ! ! ! ! ! ! m =a " b( ) p 2

+ q2( )

2 2 q2!

NAI5.2.A.)!.(25!23.,!:-!

! p2

+ q2( )2

a ! b( )2

! 4aq2

a ! b( ) p 2

+ q2( ) + 4a2q 2

p2

+ q2( ) ! 2 p2 q4 = 0 !

! ! ! a2

p2

+ q2( )

2

+ b2

p2

+ q2( )

2

" 2ab p2

" q2( ) p2 + q2( ) " 2 p2q 4

= 0 !

!

!

Page 6: ABCD BC CD - WASAN · 2016. 8. 21. · !!!!a 4 E 2 + 2 ah (a 2! q 2)EF + a 2 b 2 F 2! a 2 c2 p 2 q 2 = 0 ! &'()!*+ !!!h 2 A = a 2 B = a 2 (p 2! a 2)+!!!!! a 4 E 2 + 2a 2 DEF + a 2

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !"#"!! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !"#$!

Formula!"#! %&'!()*(+,-! O1! ./0! O

2! !

)/-(*)1,!23,!,++)4-,!.2!,.(3!2&'!4')/2-5!

.-!-3'&/6! ! %3,!()*(+,! O3! ()*(78-(*)1,-! !

23,!()*(+,! O1! ./0!)/-(*)1,-!23,!()*(+,!

O2! .2!23,!-.8,!4')/2! P ! '9!2./:,/(;6!

! ! <,2! p, q, d1, d

2! ./0! d

3! 1,!23,!-,8)=8.>'*5!8)/'*!.?,-5! !

23,!0).8,2,*-!'9!23,!()*(+,-! O1, O

2! ./0! O

36!

! ! d1

2d2

2p4! 2d

1d2

3p4+ d

2

4p4+ 4d

3

2q6!

! ! ! ! ! ! ! ! ! ! ! ! !4d2d3p2q4+ 4d

1d2

2d3p2q2! 0 !

@ProofA! B;!C'*87+.!DEE5!23,!(''*0)/.2,-!'9! O 3 x 3, y 3( ) ! F!

! !x3= !

d2p2

! d3q2( ) q

2

! d2

2( )

2d2q p 2

! q 2( )!

! ! y3=

d2! d

3( ) d

2

2

p2

! q4( )

2d2

p2 ! q2( )!

G'*,'H,*5! x =(''*0)/.2,-!'9! O1! F!

! ! x1=

p2

! q2( ) q 2

! d1

2( )2q

!

I/!JO1O3H2K! ! !

d1+ d

3( )

2

4= x

1! x

3( )

2

+ y3

2

!

! !

d1

+ d3

( )2

4=

p2

! q2( ) q

2

! d1

2( )

2q p2 ! q 2( )+

d2p2

! d3q2( ) q

2

! d2

2( )

2d2q p2 ! q2( )

"

#

$ $

%

&

' '

2

!

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! +d2! d

3( ) d

2

2

p2

! q4( )

2d2

p 2 ! q 2( )

"

# $

% $

&

' $

( $

2

!

!

! ! ! d22

q2d1 + d3( )

2

p2! q

2( )

= d2 p2 ! q2( ) q

2 ! d12( ) + d2 p

2 ! d3q2( ) q

2 ! d22( )"

# $

% & '

2

!

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! +q2

d2! d

3( )

2

d2

2

p2

! q4( ) !

! d2!2q

2

+ d1

2

+ d2

2( )p2 + 2d 3q2

d1d2+ q

2( ) !

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! = 2 d2p2

! d3q2( ) q

2

! d1

2( ) q 2

! d2

2( ) !

! ! ! d2

2

!2q2

+ d1

2

+ d2

2( )2

p4

+ 4d3

2

q4

d1d2+ q

2( )2

!

! ! ! ! ! ! ! +4d2d3q2

!2q2

+ d1

2

+ d2

2( ) d1d2 + q2( )p2 !

! ! ! ! ! ! ! ! ! ! ! ! !4 d2p2

! d3q2( )2

q2

! d1

2( ) q2 ! d22( ) = 0 !

! ! ! ! ! d2

2

d1" d

2( )

2

p4

+ 4d2d3q2

d1d2" q

2( )p2 + 4d 32

q6

= 0 !

!

!

Page 7: ABCD BC CD - WASAN · 2016. 8. 21. · !!!!a 4 E 2 + 2 ah (a 2! q 2)EF + a 2 b 2 F 2! a 2 c2 p 2 q 2 = 0 ! &'()!*+ !!!h 2 A = a 2 B = a 2 (p 2! a 2)+!!!!! a 4 E 2 + 2a 2 DEF + a 2

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !"##!! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !"#$!

Formula!"#! %&'()*+,!-&).&&,!/+'0)*+,! !

! ! ! ! ! ! ! ! ! ! ! ! (,1!2+&33*2*&,)!+3!40(15()*2!&40()*+,!

6&)! a7 0, b7 0, c708 b2!4ac9 0 ! *,!(!40(15()*2!&40()*+,

a + bx + cx2

= 0 .!

:! 6&)! ! and ! be the /+'0)*+,/!*,!(!40(15()*2!&40()*+,! !

! ! a ! bx + cx2

= 0 ,!

! ! ;! !b + c " + #( ) = 0 ! ! <! a ! c"# = 0 ! ! !

! ! ! !=! a ! + "( ) # b!" = 0

>?@(AB'&C! D,!)E&!3*F05&!GH!'&)! d1, d

2, d

3, x

1! !

! ! ! (,1! x2! -&!1*(A&)&5!+3!&(2E!2*52'&! O1, O2, O3 ,O4 ! !

! ! ! (,1! O5 I! !

! ! ! ! ! JE&!5&'()*+,/!+3! x2! +5! x

2! -&2+A&!/KAA&)5*2('!&@B5&//*+,I!

! ! ! ! ! (,1!+,&!+3! x ! */! !H!)E&!+)E&5! x ! */! ! I!

L! 6&)! ! and !" be the /+'0)*+,/!*,!(!40(15()*2!&40()*+,! !

! ! a + bx ! cx2

= 0 ,!

! ! ;! !b + c " ! #( ) = 0 ! ! ! ! ! <! a ! c"# = 0 !

! ! =! a ! " #( ) " b!# = 0 !

M! 6&)! ! and !" be the /+'0)*+,/!*,!(!40(15()*2!&40()*+,! !

! ! a ! bx ! cx2

= 0 ,!

! ! ! ;! b + c ! " #( ) = 0 ! ! <! a ! c"# = 0 !

! ! ! =! a ! " #( ) + b!# = 0 !

! >?@(AB'&C! D,!)E&!3*F05&!NH!'&)! x1, d2 , d3 , d4! !

! ! ! (,1! x5 ! -&!1*(A&)&5!+3!&(2E!2*52'&! O1, O2 ,O3, O4 ! !

! ! ! (,1! O5 I! !

! ! ! ! ! !OE&,!.&!3*,1!)E&!5&'()*+,/!-&).&&,!)E&/&!(,1!

! ! ! ! ! )E&K!-&2+A&!/KAA&)5*2('!&@B5&//*+,!+3 x1 or H! x5 !

! ! ! (,1!+,&!+3! x! */! !H!)E&!+)E&5! x! */! ! I!

!

! !

!

Formula!"$! ! ! ! ! ! ! P&5*&/! ! QGR

:! S1= 1 + 2 + 3 + L + n =

1

2!n n + 1( ) !

L! S2= 1

2

+ 22

+ 32

+ L + n2

=1

3!n n + 1( ) n + 2( ) !

M! S3= 1

3

+ 23

+ 33

+ L + n3

=1

4!n n + 1( ) n + 2( ) n + 3( ) !

S! S4= 1

4

+ 24

+ 34

+L + n4

=1

5!n n +1( ) n + 2( ) n + 3( ) n + 4( ) !

T! S5= 1

5

+ 25

+ 35

+L + n5

=1

6!n n +1( ) n + 2( ) n + 3( ) n + 4( ) n + 5( )

U S6= 1

6

+ 26

+ 36

+ L + n6

=1

7!n n + 1( ) n + 2( ) n + 3( ) n + 4( ) n + 5( ) n + 6( )

! ! ! ! ! V! ! V! ! V! ! ! ! ! ! ! ! ! ! ! ! !

! ! ! ! ! ! >The Proof is omitted.C

Formula!"%& & & & & & & & P&5*&/! ! QNR&

& ! WK!3+5A0'(!GXYH!

:! n2

+ n = 2 S1!

L! 2n3

+ 3n2

+ n = 6 S2!

M! n4

+ 2n3

+ n2

= 4S3!

S! 6n5

+ 15n4

+ 10n3

! n = 30S4!

T! 2n6

+ 6n5

+ 5n4

! n2

= 12S5!

U! 6n7

+ 21n6

+ 21n5

! 7n3

+ n = 42S6!

Z! 3n8

+12n7

+14n6

! 7n4

+ 2n2

= 24S7!

! ! ! ! ! V! ! V! ! V

! ! ! ! ! ! >The Proof is omitted.C!

Page 8: ABCD BC CD - WASAN · 2016. 8. 21. · !!!!a 4 E 2 + 2 ah (a 2! q 2)EF + a 2 b 2 F 2! a 2 c2 p 2 q 2 = 0 ! &'()!*+ !!!h 2 A = a 2 B = a 2 (p 2! a 2)+!!!!! a 4 E 2 + 2a 2 DEF + a 2

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !"#$!! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !"#%!

Example!! &'(!)(*+,*-./0!-1()!/2! !

34/!5/6708(63!(99+:)()!/03'/7/6-9+;(! !

-6<!+6)50+=(!3'(!9-07(!(99+:)(>!-)!)'/46?!

! ! ! ! @(3! p, q ! -6<! m! =(!3'(!)(*+,*-./0>! !

*+6/0!-1()!/2!3'(!9-07(!(99+:)(!-6<!3'(!)(*+,*-./0!-1+)!/2!

! ! 3'(!)*-99!(99+:)(?! ! Find the minor axes n of the small ellipse.

AB6)?C! D6!3'(!2+780(!E>!=F!2/0*89-!GH>!

! ! ! ! ! ! ! ! ! ! AB2

=1

2m2+ n

2( ) ! IJ!

! KF!2/0*89-!GE>! CD2

=m2n2

m2+ n

2!

! ! ! ! ! DE2

= 2CD2

=2m

2n2

m2+ n

2! IL!

&'(!2+780(!M!+)!3'(!2+780(!3'-3!(1:-6<(<! !

3/!p

q! 3+*()!3/!-!*+6/0!-1+)!<+0(53+/6! !

/2!3'(!9-07(!(99+:)(!+6!3'(!2+780(!E?!

&'(0(2/0(>! ! ! ! ! GH =p

qAB!

KF!2/0*89-!NG>! MN2

= AB2

+ GH2!

" # $ % p

2=p2+ q

2

q2

AB2

! p2! IO!

P(65( IJ =p

qKL ! IIQ! ! IJ

2

+ KL2

=p2 + q2( )q2

DE2

! ! IIR!

KF!2/0*89-!GST! IJ2

+ KL2!

" # $ p

2MN

2

% p2

+MN2!

" # $ IJ

2

&KL2

= 0 !

! U8=)3+383(!QTRT! p2+ q

2( )MN2

! p2DE

2

!DE2

"MN2

=0 !

! U8=)3+383(!OT! p2+ q

2( )AB2

! p2q2! AB

2

"DE2

= 0 !

U8=)3+383(!JTLT! p2

+ q2( ) m 2

+ n2( ) ! 2 p2 q2 ! 2m 2

n2

= 0 ! IIV!

! ! ! ! ! ! ! ! ! n =2 p

2

q2

" p2

+ q2( )m 2

p2 + q 2

" 2m2

Example"! &'(!)(*+,*-./0!-1()!/2!34/!9-07(!(99+:)()! E1! -6<! !

! ! E2! -0(!:(0:(6<+589-0!(-5'!/3'(0!-6<!3'(!)*-99!(99+:)(! F ! ! ! !

! ! +6)50+=()!3'(!9-07(!(99+:)(! E! -3!34/!:/+63)>!-)!)'/46?!

! ! &'(!5+059()! O1, O

2! -6<! O

3! 3/85'>!-)! !

)'/46>!-6<! O1, O

2! -0(!580W-380(!5+059()?!

! ! X+6<!3'(!*+6/0!-1+)! n ! /2!3'(!9-07(! !

(99+:)(!+6!3(0*)!/2!3'(!*+6/0!-1+)! q ! !

/2!3'(!)*-99!(99+:)(?!

AB6)?C! D6!3'(!(99+:)(!4'(0(!)(*+,*-./0!-6<!*+6/0!-1()!-0(! a, b >!

3'(!<+-*(3(0!/2!3'(!580W-380(!5+059(!4'+5'!+6)50+=()!-3!3'(!(<7(!/2! !

3'(!)(*+,*-./0!-1+)!+)!b2

a?!

! ! @(3! p ! -6<! m ! =(!3'(!)(*+,*-./!-1()! !

/2!3'(!)*-99!(99+:)(!-6<!3'(!9-07(!(99+:)(?!

&'(!<+-*(3(0!/2!3'(!5+059(! O3! Y! !

n2

mT!

X0/*!3'(!2+780(>! m = 2 !n2

m+ n !

! ! ! ! ! ! ! ! ! m = 2n ! ! IIJ!

K(5-8)(!-!)+*+9-0!0(9-3+/6!5/6)+)3)!-=/83!3'(!)*-99!(99+:)(>!3//?!

! ! ! ! ! ! ! ! ! p = 2 q ! ! IIL!

KF!(1-*:9(!E!VT! ! p2

+ q2( ) m 2

+ n2( ) ! 2 p2 q2 ! 2m 2

n2

= 0 ! IIO!

U8=)3+383(!JTL! +63/!O>!

! ! ! ! ! ! ! ! 25n2

q2

! 8q4

! 8n4

= 0 !

! ! ! ! ! ! ! ! ! n =3 + 41( )4 2

q !

!

Page 9: ABCD BC CD - WASAN · 2016. 8. 21. · !!!!a 4 E 2 + 2 ah (a 2! q 2)EF + a 2 b 2 F 2! a 2 c2 p 2 q 2 = 0 ! &'()!*+ !!!h 2 A = a 2 B = a 2 (p 2! a 2)+!!!!! a 4 E 2 + 2a 2 DEF + a 2

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !"#$!! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !"#%!

Example!! &'()!*'+*,)-! Oii =1,2,3, 4,5( ) ! !

./!01'*1!.1)!2'34).)+!'-!)563,7! !

*'+*64-*+'8)!./!.1)!9):.!*'+*,)!392! !

.1);!'9-*+'8)!.1)!,3+<)!*'+*,)! O =!

! >).! S ! 8)!.1)!3+)3!/?!.1)!8,6)!@3+.! !

392!?'92!.1)!2'34).)+! d ! /?!.1)!)563,!*'+*,)=!

AB9-=C! >).! D ! 8)!.1)!2'34).)+!/?!.1)!,3+<)!*'+*,)7!8;!&/+46,3!D7!

! ! OO3 =2 + 0.8

2dE! ! HO =

2 + 0.8( ) 2 + 0.8

8 0.8d !

&+/4!.1)!?'<6+)7!

! ! ! D = 2OO 3 + d =OO3 = 1 + 2 + 0.8( )d !F9!*'+*,)! O

37!.1)!3+)3! s ! /?!.1)!-)*./+! !

'9*,62'9<!@/'9.!B!G! ! ! s = !d2

4"7

10=7

40!d

2

!

H/+)/()+7!.1)!3+)3! s5! /?!.1)!+)<6,3+! !

@)9.3</9! O1O

2O3O

4O

5! G! !

! ! ! s5 = 5 !1

2!OH !O3O4 =

5 2 + 0.8( ) 2 + 0.8

16 0.8d2!

I1)+)?/+)7! ! S = !D2

4" 5s " s

5!

J'9*)! D = 1 + 2 + 0.8! "

# $ dE!

! ! S =

! 1 + 2 + 0.8" #

$ %

2

4d2

&7

8!d

2

&5 2 + 0.8( ) 2 + 0.8

16 0.8d2

!

! d = 40.8S

2 0.8 "1 + 2 0.8 + 4 2 + 0.8# $

% & ' " 5 2 + 0.8( ) 2 + 0.8

!

Appendix"! I1+))!-@1)+)-! S1, S

2, S

3! *'+*64-*+'8)!)3*1!/.1)+! ! !

392!'9-*+'8)!.1)!@',,3+=#I1)!@,39)!.13.!'-!432)!8;!*/99)*.'9<! !

*)9.)+-!/?!-@1)+)-!'-!@3+3,,),!./!.1)!8/../4!/?!.1)!@',,3+=!

! ! I1)!@',,3+!'-!*6.!8;!.1)!6@@)+!392!,/0)+! !

@,39)-!01'*1!*'+*64-*+'8)2!.1)!.1+))! !

-@1)+)-7!3-!-1/09=!

! ! ! &'92!.1)!-)4'K43L/+!3:'-! AB! '9!.)+4-! !

/?!.1)!2'34).)+-! d1, d

2, d

3! /?!.1)!-@1)+)-! !

S1, S

2, S

3=!

! ! ! H+=!J3./M-!56)-.'/9!'9!N*1'</!@+)?=!

AB9-=C! F9!.1)!?'<6+)!O7!,).! d ! .1)!2'34).)+!/?!.1)!83-)=!

! P;!&/+46,3!QQ7!

! ! d1

2

d2

2

d3

2

+ 2 d1+ d

2( )dd

1d2d3

2

!

! ! +2dd1

2

d2

2

d3+ d

2

d1

2

d2

2

+ d1 ! d2( )2

d2d3

2

!

! ! ! ! !2 d1 + d2( )d 2d1d2d3 = 0 ! ! RS!

F9!TACBE,).! p ! .1)!-)4'K43L/+!3:'-! !

/?!.1)!),,'@-)7!

! ! BC2

= p2!d

2= t

2

! 1( )d 2

= td2

! ! RRU!

J'9*)7!TACBVTS1S3G3E!

! ! ! ! BC: d =d1 !d3

2: G 3S3 !

! ! G 3S3

2

=d2d1 "d3( )

2

4 BC2

!

! ! ! ! ! ! =d1! d

3( )

2

4 t2!1( )

! ! RRW!

Page 10: ABCD BC CD - WASAN · 2016. 8. 21. · !!!!a 4 E 2 + 2 ah (a 2! q 2)EF + a 2 b 2 F 2! a 2 c2 p 2 q 2 = 0 ! &'()!*+ !!!h 2 A = a 2 B = a 2 (p 2! a 2)+!!!!! a 4 E 2 + 2a 2 DEF + a 2

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !"#$!! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !"%&!

'()(*+,*-.! K 2S3

2

=d2 !d3( )

2

4 t2!1( )

! ! //0!

! ! ! ! ! ! G 2S2

2

=d1 !d2( )

2

4 t2!1( )! ! //1!

23!456!7(89,6!:.! S1S2

2

=d1

2+d2

2

!

" # $

% & 2

'd1

2'd2

2

!

" # $

% & 2

!

! ! ! ! ! ! ! ! ! = d1d2! ! //;!

! '()(*+,*-.! S2S3

2

= d2d3 ! ! //<!

! ! ! ! ! ! S1S3

2

= d1d3 ! ! //=!

23!>S1S2E?!

ES

2

2

= S1S2

2

! ES1

2

!

@63A6! G 2S2 = ES1 ! (3!456!7(89,6!B!+3C!:.!D9ED4(4946!F.!;.!

! ! ! ! ! ! ! ! ES2

2

= d1d2 !d1 !d2( )

2

4 t2!1( )!

! ! ! ! t2"1( )ES2

2

= d1d2 t2"d1 + d2( )

2

4! ! //G!

! '()(*+,*-.>S1S3DH?! t

2!1( )DS3

2

= d1d3 t2!d1 + d3( )

2

4! /I!

! 23!>S2CS

3?! t

2!1( )CS3

2

= d2d3 t2!d2 + d3( )

2

4! ! //J!

! KL,6LM6,.! ! ES2

2

!CS32

!DS32

= 2CS3 "DS3 !

'9ED4(4946!G?I?J! +3C!+,,+386.!

! d1d2 t2! d1 + d2( )d3 t

2+!d1d2 + d1 + d2( )d3 + d3

2

2= 2CS3 "DS3 t

2!1( ) !

N644(38! A = d1+ d

2( )d

3! d

1d2?!

! ! ! ! ! !At2+A + d3

2

2=2CS3 "DS3 t

2!1( ) !

'9ED4(4946!I?J! +3C!+,,+386.!

! ! A2

t4

! At2

A + d3

2( ) +A + d

3

2( )2

4

!

! ! ! ! ! ! ! ! = 4 d1d3t2

!d1

+ d3

( )2

4

" # $

% & ' d2d3t2

!d2

+ d3

( )2

4

" # $

% & ' !

! A2

t4

! At2

A + d3

2( ) +A + d

3

2( )2

4!

! ! ! ! ! ! ! ! ! = 4 d1d3t2

!d1

+ d3

( )2

4

" # $

% & ' d2d3t2

!d2

+ d3

( )2

4

" # $

% & ' !

! A2

t4

! A2

t2

! d3

2

At2

+A2

4+d3

2A

2+d3

4

4! 4d

1d2d3

2

t4

!

! ! ! ! ! +d1d3d2+ d

3( )

2

t2

+ d2d3d1+ d

3( )

2

t2

!d1+ d

3( )

2

d2+ d

3( )

2

4= 0 !

! ! A2

! 4d1d2d3

2( )t 4 ! A2

t2

! d3

2

At2

+ d1d3d2+ d

3( )

2

t2

+ d2d3d1+ d

3( )

2

t2

!

! ! ! ! ! ! ! ! ! ! +A2

4+d3

2A

2+d3

4

4!

d1+ d

3( )

2

d2+ d

3( )

2

4= 0 !

N64! B = d3d1+ d

2( ) + d

1d2?C = d

1d2d3d1+ d

2+ d

3( ) ?!

! ! ! ! B2

t4

! 4C2

t4

! B2

t2

+ 5C2

t2

! C2

= 0 !

! ! ! B2

t2

t2

!1( ) ! 4C 2

t2

t2

!1( ) + C 2

t2

! 1( ) = 0 !

! ! ! ! B2

t2

t2

!1( ) ! 4C 2

t2

t2

!1( ) + C 2

t2

! 1( ) = 0 !

! ! ! ! ! ! ! ! B2

! 4C2( )t 2 + C 2

= 0 !

! ! ! ! ! ! ! ! ! B + 2C( ) 2C ! B( )t 2 + C 2

= 0 !

! ! ! ! ! ! ! ! ! ! t2

=C2

B + 2C( ) 2C " B( )!

! ! ! ! ! ! ! ! ! p2

=d 2C 2

B + 2C( ) 2C " B( )!

! ! ! ! ! ! ! p =dC

B + 2C( ) 2C " B( )! ! //O!

P6!A+3!LE4+(3! d ! 7,L)!Q! +3C!(7!R6!D9ED4(4946! d ! 7L,!O.! !

R6!A+3!LE4+(3! ABS

Page 11: ABCD BC CD - WASAN · 2016. 8. 21. · !!!!a 4 E 2 + 2 ah (a 2! q 2)EF + a 2 b 2 F 2! a 2 c2 p 2 q 2 = 0 ! &'()!*+ !!!h 2 A = a 2 B = a 2 (p 2! a 2)+!!!!! a 4 E 2 + 2a 2 DEF + a 2

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !"#"!! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !"#$!

Appendix!! %!&'())!&*+(,-!(./!(!&-012,"3.&0,34-!15-!)(,6-! !

&*+(,-7!(&!&528.9! ! !

! ! :-1! P ! 4-!15-!;23.1!2<!1(.6-.0=!2<!

15-!&'())!&*+(,-!(./!(.!(,0! AC! (./! !

AD=1! >!"#?9!

! ! @3./!15-!&3/-!2<!15-!&'())!&*+(,-!85305! !

'(A3'3B-&!15-!(,-(!2<!15-!4)+-!&-012,9!

! ! ! C,9!D2&532E(F&!*+-&132.!3.!G+&5+!;,-<9! !

H%.&9I! J5-.! DP ! (./! PQ! 4-02'-!(!&1,(36517!15-!(,-(!2<!15-!4)+-! !

! ! &-012,!4-02'-&!15-!'(A3'+'9!

:-1! AB= aK PQ= xK!

L-.0-!M FQP N BRQ K! BQ= FP = CH !

@,2'!15-!<36+,-7!MDPGOM DQHK!

! ! ! ! ! a : GP = a + x( ): a ! !

! ! ! ! ! ! GP =a2

a + x! ! PPQ!

R.!M DQH K! a + x( )2

= a2+ DH

2

! ! ! !

L-.0- DH = GP 7!S+4&131+1-!QK! !

! ! ! ! ! a + x( )4

! a2

a + x( )2

! a4

= 0 !

! ! ! ! ! ! ! a + x( )

2

=1 + 5

2a2

!

! ! ! ! !

! x = "a + a

1 + 5

2!

S+4&131+1-! a = 1K! ! x = !1 +

1 + 5

2T 0.2720196 LU!"#V!

! >.21-?! ! !"#!3&!15-!2)/!W(;(.-&-!'-(&+,-!(./!X!!"#!T! Y9Y!Z9

!

Appendix#! [5-!&-'3\'(]2,!(A3&!2<!15-!-))3;&-!

! ! 3&!2.!15-!/3(62.()!2<!15-!,-01(.6)-9!

! :-1! BE = 2U!"#VKCE = 6U!"#VK!

CF = 3U!"#V9!

! ! R.!153&!0(&-7!<3./DF9!

! ! ! C,9!R'(3B+'3F&!*+-&132.!3.!^53E+B-.!;,-<9!

H%.&9I! G=!@2,'+)(!_`K!

! ! ! ! ! ! ! ! ! BE !CF = CE !DF !

! ! ! ! ! ! ! ! ! DF =BE "CF

CE!

S+4&131+1- BE= 2K CE= 6K CF = 3K!

! ! ! ! ! ! ! DF = 1U!"#V!

!

! >.21-? !"#!3&!15-!2)/!W(;(.-&-!'-(&+,-!(./!X!!"#!T! Y9Y!Z9

Page 12: ABCD BC CD - WASAN · 2016. 8. 21. · !!!!a 4 E 2 + 2 ah (a 2! q 2)EF + a 2 b 2 F 2! a 2 c2 p 2 q 2 = 0 ! &'()!*+ !!!h 2 A = a 2 B = a 2 (p 2! a 2)+!!!!! a 4 E 2 + 2a 2 DEF + a 2

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !"#$!! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !"##!

Appendix!! $%! &'())!&*+,-,&!./-.0'&.-/1,!(!)(-2,!&*+,-,3! !

(45!(!&'())!&*+,-,!./-.0'&.-/1,&!6+,!76+,-!8!&'())!&*+,-,&3! !

(&!&+794:!

! ! ! !;+,4!6+,!5/(',6,-!7<!6+,!&'())! !

&*+,-,!/&!=>?!@!"#A3!</45!6+,!5/(',6,-! !

7<!6+,!)(-2,!&*+,-,3!

! ! B-:!CD(10E&!F0,&6/74!/4!G+/H027!*-,<:!

!

IJ4&:K! L+,!</20-,!M!/&!(!.06!.06!1D!6+,!*)(4,!6+(6!*(&&,&!6+,!.,46,-&! !

! ! 7<!6+,!1)0,!&*+,-,&!/4!6+,!</20-,!N:!

! ! ! ! O,6! D ! (45! d ! 1,!,(.+!5/(',6,-!7<! !

! ! 6+,!)(-2,!(45!&'())!&*+,-,!/4!6+,!</20-,:!

! ! ! ! ! !d

2=D + d

2sin18 ° !

! ! ! ! ! ! D =1 " sin 18°

sin18°d !

P/4.,! sin18 ° =5 !1

4Q!

! ! ! ! !D =

5 ! 5

5 !1d = 5d

!

P01&6/606,! d = 305Q!

! ! ! ! ! ! d = 682. 0007LR!"#S!

!

!

@476,A !"#!/&!6+,!7)5!T(*(4,&,!',(&0-,! !

! ! ! ! ! (45!N!!"#!U! =:=!V:

Appendix"! W6!5-(9&!(!&6-(/2+6!)/4,!<-7'!,(.+!X,-6,Y!7<!6+,! !

! ! -,20)(-!4Z274!(45!5-(9&!&'())!-,20)(-!4Z274!/46,-4())D3! !

(&!&+794:! ! [(.+!,))/*&,!/4&.-/1,&!

! ! ! 6+,!6-/(42),!(45!,(.+!&,'/Z'(\7-! !

! ! ! (Y/&!7<!6+,!,))/*&,&!/&!*(-()),)!67!,(.+! !

! ! ! &/5,!7<!6+,!)(-2,!-,20)(-!4Z274:!

! ! ! ! O,6! a, b, p ! (45! q! 1,!,(.+!&/5,!7<!6+,!)(-2,!(45!&'())! ! ! !

! ! -,20)(-!4Z2743!&,'/Z'(\7-!(45!'/47-!(Y,&!7<!6+,!,))/*&,:!

! ! ! ! ]/45! p ! /4!6,-'&!7<! a, b ! (45! q :!

! ! ! ! B-:!L&017D('(E&!F0,&6/74!/4!^(9(.+/!*-,<:!

!

IJ4&:K! W4!6+,!</20-,!N3 BC= aQ AF= bQ!

O,6! CO= RQ EO = rQ!

n= "Q!

! ! a = 2Rsin! ! ! R =a

2sin "! __`!

! ! b = 2r tan! ! ! r =b

2 tan "! __a!

W4!bCOE Q!

! CE2

= CO2

!EO2

= R2! r

2! ! __c!

! ! ! AC= DF = CE + AE= R2" r

2+b

2!

! ! ! ! AB= CF = CE !EF = R2! r

2!b

2!

W4!6+,!</20-,!MQ 2CH !BC = BC2

+ AC2

" AB2

! !

! ! 2a CH "a

2

#

$ % &

' ( = R

2" r

2+b

2

#

$ % &

' ( 2

" R2" r

2"b

2

#

$ % &

' ( 2

!

! ! ! ! ! ! ! CH =a

2+b R

2" r

2

a!

L+,-,<7-,3!),6 AH = hQ!

Page 13: ABCD BC CD - WASAN · 2016. 8. 21. · !!!!a 4 E 2 + 2 ah (a 2! q 2)EF + a 2 b 2 F 2! a 2 c2 p 2 q 2 = 0 ! &'()!*+ !!!h 2 A = a 2 B = a 2 (p 2! a 2)+!!!!! a 4 E 2 + 2a 2 DEF + a 2

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !"#$!! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !"#%!

! ! h= AC2

!CH2

= R2! r

2+b

2

"

# $ %

& ' 2

!a

2+b R

2 ! r 2

a

"

# $

%

& '

2

!

&'()*+*'*,!-./.!

h = R2

! r2

+b2

4!a2

4!b2

R2

! r2( )

a2

=a2

4 tan2 "

!2b

2

4 tan2 "

+b4

4a2tan

2 "!

! ! ! =1

4 tan2 !

a2

" 2b2

+b4

a2

#

$

%

& =

1

2 tan!a "

b2

a

#

$

%

& !

&'()*+*'*,!-0/! +1*2!3.!

! ! CE2

= R2! r

2=

a2

4 sin2 "

!b2

4 tan2 "!

a2

4 sin2 !

=a2

4 1 " cos2 !( )

=a2

1 + tan2

!( )4 tan

2 !=

a2

4 tan2 !

+a2

40!

! ! ! ! ! ! ! CE2

=a 2

4tan2q

+a 2

4"

b2

4tan2q!

45!6278'9:!;<.!

! ! ! q2AC

2

! AB2"

# $ %

2

h !q( )!a 4 2 h !q( )2

h !q( )+ a2hp

22 h !q( )

2

!0 !

! ! AC2

! AB2"

# $ %

2

= R2! r

2+b

2

"

# & $

% ' 2

! R2! r

2!b

2

"

# & $

% ' 2(

)

*

+

,

-

2

!

! ! ! ! ! ! ! ! = 4b2

R2

! r2( ) = b

2

a2

+a2

tan2 "

!b2

tan2 "

#

$

%

& !

&'()*+*'*,!*=+).!

! ! ! b2

q2

a2

+a2

tan2 !

"b2

tan2 !

#

$

%

& h " q( ) !

! ! ! ! ! ! ! ! ! ! !a4

2h ! q( )2

h ! q( ) + a 2

hp2

2h ! q( )2

! 0 !

! ! p =a2 h " q( )

h"

b2 q2 h " q( )

a2

h 2h " q( )2

a2

+a2

tan2 #

"b2

tan2 #

$

%

&

' !

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! =2>,?,70h =1

2 tan! na "

b2

a

#

$

%

& !

!

Appendix!! @!:785!A,B+A,A!*2!B21)'9*!>+*=!4!:785!:1A!*2! !

:**:BC!*=,!B:)*9,!2D!E!:785F! ! 4!+)!+1!*=,!A',!127*=!2D!*=,! !

B:)*9,!E.!:1A!@!+)!+1!*=,!)2'*=>,)*!2D!*=,!B:)*9,!2D!EF!

! ! ! G=,!A+)*:1B,!2D!4!:1A!E!+)!*>2!H!"I!921J,7!*=:1!*=,!A+)*:1B,! !

! ! 2D!@!:1A!*=,!B:)*9,!EF! ! @!)*7:+J=*!9+1,!A+)*:1B,!2D!@!:1A!4!+)! !

21

7! *+8,)!*=,!A+)*:1B,!2D!@!:1A!*=,!B:)*9,!EF!

! ! ! ! @*!*=+)!*+8,.!D+1A!:!)*7:+J=*!9+1,!A+)*:1B,!2D!@!:1A!4F!

! ! ! ! ! ! ! K7F!G212C+L)!M',)*+21!+1!@)='!N7,DF!

O@1)FP! Q,*! BO= x0AB

BO=b .!:1A!@!+)! a ! H!"I!D728!*=,!A+)*:1B,!2D!4! !

! ! :1A!*=,!B:)*9,!E!:>:5F!

! ! ! bx =x

2

!

" # $

% & 2

+x

2+ x + a

!

" # $

% & 2

!

! ! b2

x2

=x2

2+1 + 2( )

2

2x2

+ 2 1 + 2( )ax + a 2

!

! ! ! b2

x2

= 2 + 2( )x 2 + 2 1 + 2( )ax + a2 !! ! ! 2 + 2 ! b

2( )x 2 + 2 + 2( )ax + a2 = 0 !

! ! ! x =a

2 2 + 2 " b2( )

" 2 + 2( ) " 4b2

" 2{ } !

! ! ! ! ! ! AB = bx =ab " 2 + 2( ) " 4b

2" 2{ }

2 2 + 2 " b2( )

!

&'()*+*'*,! a = 20b = 21

7.!

! ! ! ! ! ! ! ABR 13.574325LS!"T!

! H12*,I !"!+)!*=,!29A!U:N:1,),!8,:)'7,!:1A!V!!"!R! W XF

!

!

! H12*,I! ! YZ!+)!*=,!'1+*!2D!*=,!U:N:1,),!29A!9,1J*=.!V!YZRWXF!

Page 14: ABCD BC CD - WASAN · 2016. 8. 21. · !!!!a 4 E 2 + 2 ah (a 2! q 2)EF + a 2 b 2 F 2! a 2 c2 p 2 q 2 = 0 ! &'()!*+ !!!h 2 A = a 2 B = a 2 (p 2! a 2)+!!!!! a 4 E 2 + 2a 2 DEF + a 2

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !"#$!! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !"#%!

Appendix!! &'(!)*+,-./0,*!12*,3+4!+5!,!*(./0,*!)()*,'(4*6-7!

&'*((!51'(*(5! S1, S

2! ,-4! S

3! 8+*8/358*+9(! !

(,8'!6)'(*:!)6/8'!)'(!9,5(!6;! !

)'(!12*,3+4!,-4!)6/8'!(,8'!5+4(!6;! !

)'(!12*,3+4!,)!6-(!16+-)7!

! ! <()! d1, d

2! ,-4! d

3! 9(!)'(!4+,3()(*5!6;! !

)'(!51'(*(5! S1, S

2! ,-4! S

37! ! =+-4!)'(!5+4(!6;!)'(!*(./0,*! !

)()*,'(4*6-!+-!)(*35!6;! d1, d

2! ,-4! d

37!

! ! ! ! ! ! ! ! ! ! >*7!?3/*,+@5!A/(5)+6-!+-!B46!1*(;7!

CD-57E! =*63!)'(!;+./*(!FG!

! ! ! ! ! ! ! ! ! ! S1S2

2

=d1

2+d2

2

!

" # $

% & 2

'd1

2'd2

2

!

" # $

% & 2

!

! ! ! ! ! ! ! ! ! S1S2 = d1d2 !

H+3+0,*02:!0() S2S3 = d2d3 G! S1S3 = d1d3 !

s =1

2d1d2+ d

2d3+ d

1d3( )G!

I2!J(*6-@5!;6*3/0,G!

! KS1S2S3= s s ! d

1d2( ) s ! d

2d3( ) s ! d

1d3( ) !

! ! ! ! ! =1

42d

1d2d3d1+ d

2+ d

3( ) ! d

1

2

d2

2

+ d2

2

d3

2

+ d1

2

d3

2( ) !

! ! ! ! =1

44d

1d2d3d1+ d

2+ d

3( ) ! d

1d2+ d

2d3+ d

1d3

( )2

! LM!

I2!N26N/N+O/)5/

PQR S-6)(T! !

! ! ! 2 d1d2+ d

2d3+ d

1d3

( )

! ! ! ! ! ! ! ! ! ! ! +8 3!S1S2 S3 "3 # A # B 2

= 0

H/95)+)/)(!MG!

! A ! B =2

3

! d1d2 + d2d3 + d1d3 + 3 4 d1d2d3 d1 + d2 + d3( )" d1d2 + d2d3 + d1d3( )2# $ % & ' ( LU!

V-!)'(!;+./*(!W:! BH1 ! +5!69),+-(4:!

V-!KBH1S1G!

9

4d1

2

!1

4d1

2

= 4BH1

2

!

! ! ! ! ! ! ! ! BH1 =d1

2! ! LLX!

V-!KAP1P ! 6;!)'(!;+./*(!Y:!

! ! ! ! ! ! AP =2

3PP1 ! ! LLZ!

H+-8(:! ! ! ! PP1 =BH1G!

H/95)+)/)(!X! +-)6!Z:!

! ! ! ! ! ! AP =2

3d1 ! ! LL[!

H+3+0,*02:! ! CQ=2

3d2 ! ! LL\!

! ! ! ! ! PP2 + QQ2 =2

3d3 ! ! LL]!

=*63!)'(!;+./*(!YG!

! ! ! ! ! ! AC= AP+ P2Q2 + CQ+ PP2 + QQ2( ) !

H/95)+)/)(!UG[G\G]G!

! ! ! ! ! AC=2

3d1 + ! A ! B +

2

3d2 +

2

3d3!

! ! ! ! ! ! ! ! ! ! ! ! ! AC=2

3d1 + d2 + d3( )[ !

! !

Page 15: ABCD BC CD - WASAN · 2016. 8. 21. · !!!!a 4 E 2 + 2 ah (a 2! q 2)EF + a 2 b 2 F 2! a 2 c2 p 2 q 2 = 0 ! &'()!*+ !!!h 2 A = a 2 B = a 2 (p 2! a 2)+!!!!! a 4 E 2 + 2a 2 DEF + a 2

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !"#$!! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !"%&!

! ! !'()*+,! -.!/.)/0/+120*30

456 7!8! 13!)9*:1(+;7!

<(!*=+!>1?=*!@1?0>+7!A+*! D ! 9+!*=+!;1:B+*+>!)@!*=+!C1>CA+3! S1, S2 ! !

:(;! S3 D

! ! ! ! A M =3

2D ! ! EEF

! ! ! MS2 =2

D!S1S2S3 ! ! EEG

! ! ! ! " A " B " C =3

4

" A " B 2

! ! EEH

! ! ! ! A S2 =" ! A ! B ! C

D! ! EEI

J>)B!*=+!@1?0>+K ! ! A M + MS2 "! A S2 = 0

L093*1*0*+!FKGKHKI! :(;!:>>:(?+7!

! ! ! ! ! !3

2D +

2

D!S1S2 S3 "

3

4D

# A # B 2

= 0

! ! ! ! ! ! 6D2

+ 8 3"S1S2 S3 #3 $ A $ B 2

= 0

-.!4567!3093*1*0*+! D2

=1

3d1d2+ d

2d3+ d

1d3

( ) 7!:(;!>+3*)>+7!

! ! ! ! 2 d1d2 + d2d3 + d1d3( )+ 8 3!S1S2 S3 "3 # A # B 2

= 0 ! ! EE8

!

!

'()*+,!/.)/0/+120*30

456 ! 13!()*!(+C+33:>1A.!C)>>+C*D!

M=+>+@)>+7!*=)0?=!N>)O1(?!8 13!(+C+33:>.7!1*!=:3!()*!9++(! !

C)(@1>B+;!.+*!*=:*!13!C)>>+C*D!

Appendix!! M=+!P0:;>:(?A+! ABCD ! 13!

! ! ! :!*>:N+Q)1;!)@! !B =!C= 90° ! :(;! !

! ! ! *=+!N)1(*! P ! 13!)(!*=+!31;+! BCD!

! R+*! AP= aKDP = bK!

! ! J1(;!*=+!A+(?*=!)@! BC!S=1C=!B:T1B1Q+3!*=+!:>+:!)@!*=+! !

*>:N+Q)1;! ABCD D!

UV(3DW! *>:N+Q)1;! ABCD =XABP +XCDP +XAPD !

<(!XABP 7!A+*! BP = xKXABP = SK!

! ! ! ! ! ! ! ! S =1

2x a

2

! x2

! !

! ! ! ! ! ! 2dS

dx= a

2

" x2

"x2

a2" x

2

!

! R+**1(?!dS

dx= 0K!

! ! ! ! ! ! ! ! a2

! x2

=x2

a2! x

2

! !

! ! ! ! ! ! ! ! ! ! x =a

2!

<(!3=)>*7!S=+(!XABP ! 9+C)B+3!:(!13)3C+A+3!*>1:(?A+7!*=+!:>+:!)@!1*! !

9+C)B+3!B:T1B0BD!

L1B1A:>A.7!:* CP =b

27!*=+!:>+:!)@!XCDP ! 9+C)B+3!B:T1B0BD!

Y)>+)O+>7!:*! !APD = 90° 7!*=+!:>+:!)@!XAPD ! 9+C)B+3!B:T1B0BD!

! ! M=+>+@)>+7!*=+!:>+:!)@!*=+!*>:N+Q)1;! ABCD ! 9+C)B+3!B:T1B0B! !

:*!*=+!(+T*!C:3+!Z!

! ! ! ! ! ! BC=a

2+

b

2=

2 a + b( )2

!

!

Page 16: ABCD BC CD - WASAN · 2016. 8. 21. · !!!!a 4 E 2 + 2 ah (a 2! q 2)EF + a 2 b 2 F 2! a 2 c2 p 2 q 2 = 0 ! &'()!*+ !!!h 2 A = a 2 B = a 2 (p 2! a 2)+!!!!! a 4 E 2 + 2a 2 DEF + a 2

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !"#"!! ! ! ! ! ! ! ! ! ! ! !

Appendix!! $!%&'()*+,-! ABCD ! ,.! !

! ! !B =!C= 90° /! AC= a ! '0-! BD = b 1!

! 2,0-! BC! ,0!%)&3.!+4! a ! '0-! b /! !

56)0!%6)!'&)'!+4!%6)!%&'()*+,-! ABCD ! !

7)8+3).!3'9,3:31!

! ! ;&1!<,%'3:&'=.!>:).%,+0!,0!?-+!(&)41!

!

@$0.1A! B0!%6)!4,C:&)!D/!56)0! BP + DQ ! 7)8+3).!%6)!3'9,3:3/! !

! ! %6)!'&)'!+4!%&'()*+,-! ABCD! 7)8+3).!%6)!3'9,3:31!

! ! ! ! $0-!56)0!,%!,.! AC!BD /! BP + DQ ! !

! ! 7)8+3).!%6)!3'9,3:31!

! B0!%6)!4,C:&)!E/! !ECA = 90°F CE= b/!

.+!%6'%! BP + BDFBD!EC 1!

G)08)!HABC IHACE F!

! ! ! ! BC: a = b: a2+ b

2!

! ! ! BC=ab

a2+ b

2=ab a

2+ b

2

a2+ b

2!

!

!

The Original text

J6)!+&,C,0'K!%)9%!+4!%6,.!7++L!,.!.'0(+MN + N: % .:

OPQR1!

! ! B%!,.!,0!S5'.'0.6+T O UV

'&86,W)!XYZ[! ,0!%6,.!G\1!