幾 何

.. =,,, [ ] (1 1 1 = 1 = 1 > 1 ( (2 2 2 = 2 = 2 < 2 ( (1(2,,, 1 2 ~94~

(1 (2 (3 (a G (b (c G (d G O = 1 2 O O O [ ] O 1 = O 1 = 1 2 O= O = 1 O ~95~

1. 2. = 3. M M M=M M,,,, 4. 5. ( (1 (Menelaus 98 << >> (Sphaerics 1678 (eva (Menelaus = 1 [ ] / / / = / / = / / = / / = / / / / / /=1 / / / / / / (Menelaus ~96~

= 1 [ ] / / / / =1 = 1 / / = =/ (Menelaus sin sin sin sin sin sin = 1 [ ] = 1 1= = = sin sin sin sin sin sin = sin sin sin sin sin sin M N T T M N [ ] Menelaus N M ~97~ T

N N T T M M = =1 T M N (2 (G.eva 1648~1734 << >> (eva = 1 G G [ ] ( = G G G G G G =1 ( L R L Q L R Q G = R = Q Q = G G = R = Q R = Q R Q R =1 ~98~

(eva = 1 [ ] (1 (2 / / eva / =1 = 1 =/ / (3 / = / / / (eva sin sin sin sin sin sin = 1 G [ ] = 1 1= = = sin sin sin sin sin sin G ~99~

= sin sin sin sin sin sin (1992 MO O r 1 2 O 1 O 2 r 1 r 2 1 2 1 2 1 2 O O 1 2 O 2 1 [ ] OO 1 O 2 1 2 OO1 1 O 2 O 1 1 O 1 O OO 2 O 1 O 2 O 2 2 2 O = r r 1 r 1 r 2 r 2 r =1 2 1 O 2 O 1 eva O O 1 2 O 2 1 (4 O tolemy + = [ ] =(+= + = = [ ] = =.(* = =..(** (*(** = = =(+= + ~100~

tolemy +,,, [ ] = = = (* = = + = + = =.(** (*(** + = + =(+ + =180 + =180,,, (Simon ( [ ] 2= 3 1= 2 3= 4 + 2=90 = 1+ 6 = 6 2=90 =90 6=90 5= 4= 3 6 2 1 3 4 5 ~101~

(Simon [ ] 1= 2 3= 4 3= 2 1= 2= 3= 4 6=90 1=90 4= 5 + =180 M,,,Q M=MQ 6 2 1 3 4 5 [ ] M=x MQ=y M= M=a α α M QM QM M M QM QM M =1 M sinα M Q sinα M MQsinγ M M sinγ M sinβ Q M sinβ MQ M sinδ M M sinδ =1 β δ γ M γ δ β Q (MQ 2 =Q Q (M 2 = =a 2 x 2 Q Q=Q Q=a 2 y 2 (a 2 x 2 y 2 =(a 2 y 2 x 2 x 2 =y 2 x=y [ ] M=x MQ=y M= M=a G H G H M K L Q Q K L x y = G QK = H QL ~102~

x2 y 2 = G H QK QL = (G QL (H QK G QL = Q H QK = Q x2 y 2 = Q Q = Q Q = (a+x(a x (a y(a+y x=y [ ] x 2 +(y m 2 =R 2 y=k 1 x y=k 2 x µ(x 2 +(y m 2 R 2 +λ(y k 1 x(y k 2 x=0 y=0 Q (µ+λk 1 k 2 x 2 +µ(m 2 R 2 =0 x 1 x 2 x 1 +x 2 =0 M=QM y M Q x 1815 < > 1815 1973 Steven ( oxeter 1983 ( (5 H G O [ ] O O M O OM G O OM// OM= 1 2 H H H// M // H H = H H=2 OM ~103~

M OH G / G / H MG / O G/ H = MG/ MO G / = 2 MG / G=G / GH H G O GO =G MG =2 1 O 1 O 2 r,r d O 1 ( O 2 d 2 =R 2 m 2Rr( 1 d R 1 d+r = m 1 r ( O 1 + [ ] ( α α r r α O 2 O 1 α β α+β β O 2 α α β α β α O 1 β G R 2 d 2 =(R+d(R d O 1 O 2 R 2 d 2 =(R+d(R d=o 1 O 1 =O 1 O 1 O 1 O 1 = r sinα R2 d 2 =2Rr O 1 2Rsinα 2Rsinα= O 1 = [ ] ( = = =α O 1 = O 1 =β O 1 = O 1 =α+β O 1 = ~104~

2Rsinα= = O 1 O 1 = r sinα R 2 d 2 =(R+d(R d=o 1 O 1 =O 1 O 1 = r sinα 2Rsinα=2Rr ( = = =α GO 1 = O 1 =β O 1 = O 1 =α β O 1 = 2Rsinα= = O 1 O 1 = r sinα d 2 R 2 =(d+r(d R=O 1 O 1 =O 1 O 1 = r sinα 2Rsinα=2Rr N 1.,, M,N,L M,N,L ( 2. M 3. L (a =1 (b ( ( 4. 5. +=+ +=+ +=+ ~105~

6. ( Gergonne I N M 7. 3 4 1 M N M =N M N M N 8. G G= + ( tolemy 9. K M N K M M = N N 10. 1 1 1 ( 2 2 2 1 1 1 2 2 2 K M N 1 2 1 2 2 1 ~106~

SSS SS S S RHS (1 a Q Q = a G a G / G a G / G a G / Q (a (b [ ] + + + + + / / + / + ~107~

[ ] / / / / ' // // // = = >0 [ ] // // // = = 120 [ ] R Q R RS T S = = TS 120 Q R ~108~

(2 / / / / O O θ( θ θ / / O θ (a (b (c =3 =4 =5 [ ] ' ' 60 / / / / / =60 = / =3 / = ' / / =3 / = =4 =5 / / =60 +90 =150 2 = /2 + / 2 2 / / cos150 =25+12 3 = 25 + 12 3 ~109~

[ ] (a ++ (b 60 = / = / = / / ++= / / + / + (c / 60 / / / / + / + / ++= / / + / + (d (b(c / ++ / ( / 60 ++ / / ' ' 2 ' 4 / ' 3 1 [ ] (1 / / (2 / / (3 / =++ / / (SS 1= 2 / / 3= 4=60 / / / / ~110~

= / / / (SS = / / ++= / / + / + = / l 1 l 2 l 3 [ ] (1 l 1 l 2 l 3 (2 60 l 1 l / 1 l / 1 l 3 [ ] (1 l 2 l 1 60 l / 1 (2l 3 l / 1 l1 l 1 / l 1 l 2 l 3 ~111~

(3 ( / L / L L / / L / (a (b (c Q R QR [ ] (1 Q R (2 Q Q1 Q 2 Q= Q1 RQ= RQ 2 QR Q 1 + R + RQ 2 (3 Q Q 1 Q 2 Q 1 R Q 2 QR Q QR Q 2 3 1 2 4 Q1 R Q 1 R Q 2 Q Q (4 Q QR = Q+ QR+ R = Q 1 + R + RQ 2 = Q 1 Q 2 Q 1 Q 2 = 1+ 2+ 3+ 4=2( 1+ 2=2 ~112~

2 2 1 2 2 1 2 QR = Q 1 Q 2 = Q + Q Q Q cos 2 Q= Q 1 = Q 2 =l QR = 2l 2 (1 cos2 =2lsin 2lsin Q=l Q Q Q Q=l [ ] (1 Q (2 Q Q1,Q 2 (3 Q 1 Q 2 R QR Q R QR ( Q R O 1, 2, 1, 2, 1, 2 1, 1, 1,, 2, 2, 2,, Q l 3 1 1 1 1 l 2 m 2 2 O 2 2 2 Q 2 1 2 1 m 1 m 3 l 1 [ ] 1 2 O l 1 1 2 m 1 1 2 l 1 m 1 O l 2 m 2 O l 3 m 3 O l 1 l 2 l 3 m 1 m 2 m 3 Q ~113~

1. 2 1 2. H H 3. = = 4. / / / (1 / = / = / (2 / / / ( (3 = = =120 5. Q 2 + 2 + 2 + 2 = 2 + 2 +4 Q 2 (88 6. r 1 O O=r 2 r 2 >r 1 7. ( M,,,Q M=MQ [ MO / M / QM] O 8. O 1 O 2 1 2 Q 1 Q 2 1 2 Q 1 Q 2 M 1 M 2 1 Q 1 2 Q 2 O 1 O 2 = M 1 M 2 M Q ~114~

(1 (a (b (a 0 (b a 2 = a a (a ( = + = O O (b,, α,β α+β=1 O=α O+β O (c λ =λ Q R Q R [ ] = a = b =m a =n b Q R 1 = 1 = 2 2 (t a +(1 tn b = t 2 2 a + (1 tn b 1 Q = 2 ( a + b R = 1 2 (m a +n b Menelaus =1 n 1 1 m m 1 1 t t =1 mn+t mnt m=0 =x Q +y R x+y=1 ~115~

t 2 a + (1 tn x + my t = 2 2 x + ny (1 t n = 2 2 2 b =x( 1 2 ( a + b +y( 1 2 (m a +n b mn nt mnt x = m n t n + nt y = m n mn+t mnt m=0 x+y=1 =x Q +y R x+y=1 Q R NM Q QN [ ] QN =0 QN =( + ( Q + N Q = N + Q ( Q N = N + Q M N N = N cos N Q = Q cos Q N= Q cos N=cos Q QN = N + Q =0 QN ~116~

(2 (an x n =1 1 n (n x n =1 z k =cos 2kπ n +i sin2kπ n k = 0,1,2,,n 1 1 n n 1, z 1,z 2,,z n 1 1 n n (b e iθ =cosθ +isinθ (c z 1 z 2 (1 z 2 z 1 +z 2 O z 1 (2 z 1 z 2 z= z 1+z 2 2 (3 z z 1 =(z 2 z 1 r(cosθ +i sinθ = (z 2 z 1 r e iθ z 1 z 2 z 1 θ r (r>0 z z 1 z θ=arg( z z 1 z 2 z 1 (4 z 3 z 1 z 1 z 2 z 3 =λ (λ z 2 z 1 (5 z 3 z 1 z 1 z 2 z 1 z 3 z 2 z 1 z 1 z 2 z 3 z 2 1 +z 2 2 +z 2 3 =z 2 z 3 +z 3 z 1 +z 1 z 2 [ ] (1 π z 1 z 2 z 3 3 O z 1 z 1 θ z 2 arg( z 1 z 2 z 3 z 2 =arg( z 2 z 3 z 1 z 3 = π 3 z 1 z 2 = z 2 z 3 = z 3 z 1 =a z 1 z 2 = z 2 z 3 = (cos π z 3 z 2 z 1 z 3 3 + i sinπ 3 z 2 z 3 ~117~

z 1 2 +z 2 2 +z 3 2 =z 2 z 3 +z 3 z 1 +z 1 z 2 (2 z 1 2 +z 2 2 +z 3 2 =z 2 z 3 +z 3 z 1 +z 1 z 2 z 1 z 2 z 3 z 2 = z 2 z 3 z 1 z 3 arg( z 1 z 2 z 3 z 2 =arg( z 2 z 3 z 1 z 3 z 1 z 2 = z 2 z 3 = z 3 z 1 z 1 z 2 z 3 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 [ ] ( 1 i= 1 (1 i 1 = i (1 i 1 = i (1 i 1 = i (1 i 1 = i 2 = 1+ 1 2 2 2 (1 i=+ i(+ (a 2 2 (1 i=+ i(+.(b 2 2 (1 i=+ i(+..(c 2 2 (1 i=+ i(+..(d 2 2 2 2 ( 2 2 i = ( 2 2 ( 2 2 i = ( 2 2 ( 2 2 i = ( 2 2 ( 2 2 i =( 2 2 (a (b 2( 2 2 (1 i= +i( (c (b 2( 2 2 (1 i=( +i( ( 2 2 i = ( 2 2 ( 2 2 i = ( 2 2 ( 2 2 i = ( 2 2 ( 2 2 i =( 2 2 2 2 2 2 1 2 1 2 2 1 2 1 ~118~

=a, =b, =c a +b +c abc [ ] z z 1 z 2 z 3 f(z= (z z 2(z z 3 (z 1 z 2 (z 1 z 3 + (z z 1(z z 3 (z 2 z 1 (z 2 z 3 + (z z 1(z z 3 (z 3 z 1 (z 3 z 2 f(z z f(z 1 =f(z 2 =f(z 3 =1 f(z 1 (z z 2(z z 3 (z 1 z 2 (z 1 z 3 + (z z 1(z z 3 (z 2 z 1 (z 2 z 3 + (z z 1(z z 3 (z 3 z 1 (z 3 z 2 1 z + w z+w z z 2 z z 3 z 1 z 2 z 1 z 3 + z z 1 z z 3 z 2 z 1 z 2 z 3 + z z 1 z z 3 z 3 z 1 z 3 z 2 a +b +c abc 1 1. Q R = Q= π 4 = Q=π 6 R= R= π 12 (a QR= π 2 (b QR = R (1975 IMO 2. / / / / / (81 / / / 2 2 2 2 2 2 3. + + + + 4. G 1 G 2 G 3 G 1 G 2 G 3 5. G 1 K 1 G 1 ~119~ G 1 G 2 G 3

K 1 = 3 4 G 1 O (a OG 1 = 1 3 ( O+ O+ O(b OK 1 = 1 4 ( O+ O+ O+ O(c G 1 G 2 G 3 G 1 G 2 G 3 G 4 6. M N MN 7. 8. a 2 +b 2 +c 2 abc (1Heron L Q R + RQ L Q Heron Q L / Q / L R Q R [ ] S L Q S + SQ= / S+ SQ / Q= / R+ RQ= R + RQ S=R / L R / Q L R S R= QR / (2 Torricelli ~120~

Steiner Steiner Heron r + r Heron α=β=γ 120 120 (a 120 = = =120 (b 120 ++=+ [ ] (1 120 1 1 1 (1 1 = 1 = 1 (2 1 1 1 (3 1 1 1 1 60 1 1 = 1 1 = 1 1 = 1 = 1 1 1 1 1 1 1 1 60 1 1 =60 =60 1 1 = 1 =60 =120 =120 1 =60 1 1 = 1 =60 + 1 =180 1 = = =120 = = =60 1 1 1 1 ~121~

1 1 1 Q 1 Q Q 1 Q (2 Q Q 60 1 1 Q 1 1 1 ++= 1 ++ 1 1 = 1 Q+QQ 1 +Q 1 1 =Q+Q 1 + 1 Q 1 =Q+Q+Q (3 120 120 Q Q 60 Q 1 1 1 QQ 1 60 QQ 1 Q Q+Q+Q QQ 1 +Q+Q 1 1 1 =+ Q (3Schwarz ( (agnano 1715~1797 1775 (Schwarz, 1843~1921 Schwarz 6 5 4 3 (a + ~122~ 1 2

1= 2 + 3= 4 + 5= 6 (b (1,,H, 1= 3,,,H 2= 4 3 1= 3= 4= 2 4 H 1 2 (2 4+ 5+ =180 180 2 + 180 2 + =180 = 1 2 ( + =1 2 (180 = 2,,,,,,,,, 6 5 4 3 = = = 1+ = 2+ 1 2 ' '' ' G' '' I ' ' G H ~123~

(c GHI // // //( / / 180 2 // / // / // // // / = / = // GG / // // GG / = // GG / =2 2 GHI = GH G./ GG / =2 (4Sturm Lehmns << >> 1840 Lehmns Sturm Sturm Sturm Lehmns Sturm Lehmns = = [ ] G =G G > > G = = 1 2 >1 2 = = > G= < G=G G> G >G= (5Morley Morley( 1904 20 1909 << >> 1924 Morley Morley X Y Z XYZ ~124~

[ ] (1 =3α =3β =3γ X S X S XS S (2 SX SXZ= SXY=30 Z Y S S SXZ SXY XZ=XY (3 Z Y X / =X X // =X ZX / ZX ZX / =ZX=ZY YX // =ZY X / Z=ZY=YX // X / ZY=360 2 ZX 60 =360 2( 1 2 S+30 60 =240 S =240 (180 2β 2γ =60 +2(β+γ =60 +2(60 α =180 2α X / Z O X S Y X // ZYX // =180 2α X / ZY O X // O X / OZ= ZOY= YOX // =2α X / OX // =6α =3α O X / Z=ZY=YX // Z Y [ ] (1 =3α =3β =3γ Y=m Z=n = =c =b 3α+3β+3γ=180 α+β+γ=60 α+β=60 γ n Z sinβ = c c sinβ n= sin(α+β sin(α+β = c sinβ sin(60 γ m= b sinγ sin(60 β b c = sin3β sin3γ m n = sin3β sinγ sin(60 γ sin3γ sinβ sin(60 β sin3θ=4sinθ sin(60 +θsin(60 θ c n Z m X Y b ~125~ a

m n =sin(60 +β sin(60 +γ (2 ZY=60 +β YZ=60 +γ α+β+γ=60 60 +β 60 +γ α / / / sin(60 +β α sin(60 +γ =m n YZ=α YZ= / / / ZY=60 +β YZ=60 +α ZX=60 +α Z=180 (α+β=180 (60 γ=120 +γ YZ+ Z+ ZX=60 +β+120 +γ+60 +α=300.. XZY=60 XYZ= YXZ=60 XYZ ~126~

1.,,, ( (87 2. O O (a (b (87 3. O M O M M M (87 4. O M M O 1 G H G M H M GM 1 MH = 1 M 1 M (88 5. = =10 =40 (88 ns 20 6. = =30 =2 =4 (89 7. = =60 = =30 = ns 20 (89 8. + = + (89 ~127~

9. G O. G. T T T = (90 G O T 10. (90 11. O KO 10 MON 30 TO L O OK, OM, ON L,, O Q O O OQ O Q (90 N Q T M K L 12. L Q R (a = M M (90 (b QR M 13. =90 = O (a O (b 2 2 + 2 (90 14. L Q R QR M (89 ( 90 15. O O= O ~128~

O O H O G G// H (88 16. = 40 o = 60 o, = 40 o = 70 o G G (90 17. K 1 K = 1 K + 1 K (90 (1< > (2< > (3 < > < > (4 (5What is Mathematics Richard ourant,herbert Robbins (6 ~129~