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"a3:=0:b3:=b3:c3:=0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "a4:=a:b4:=b4:c4:=0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "a5:=a:b5:=b5:c5:=0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "# Set of equations" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "Lambda1:=factor(subs(a=a1,b=b1,c=c1,A=A1,B=B1,C=C1,R= R1,Lambda)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "Lambda2:=fa ctor(subs(a=a2,b=b2,c=c2,A=A2,B=B2,C=C2,R=R2,Lambda)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "Lambda3:=factor(subs(a=a3,b=b3,c=c3 ,A=A3,B=B3,C=C3,R=R3,Lambda)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "Lambda4:=factor(subs(a=a4,b=b4,c=c4,A=A4,B=B4,C=C4,R=R4,Lambda )):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "Lambda5:=factor(subs (a=a5,b=b5,c=c5,A=A5,B=B5,C=C5,R=R5,Lambda)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "N:=e0^2+e1^2+e2^2+e3^2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Phi:=e0*f0+e1*f1+e2*f2+e3*f3:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Delt a21:=Lambda2-Lambda1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "De lta31:=Lambda3-Lambda1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 " Delta41:=Lambda4-Lambda1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Delta51:=Lambda5-Lambda1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "# Item 1" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "e3:=solve(a*e0*e1-a*e2*e3-e0 *e1-e2*e3,e3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "*,I#e0G6\"\"\"\"I#e1GF $F%,&I\"aGF$F%!\"\"F%F%I#e2GF$F),&F(F%F%F%F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "# (a) T" }{MPLTEXT 1 0 46 "riples m1,m4,m5 and M1, M4,M5 are not congruent" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 40 "solve(\{Phi,Delta41,Delta51\},\{f1,f2,f3\}):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "assign(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "G2:=numer(simplify(Delta21)):nops(%);degr ee(G2,\{e0,e1,e2,e3\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"$l)" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "G3:=numer(simplify(Delta31)):nops(%);degree(G3,\{e0,e 1,e2,e3\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"$l)" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "H2:= factor(resultant(G2,N,e0)):nops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 " \"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "H3:=factor(resulta nt(G3,N,e0)):nops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "K2:=op(1,H2/64/e2^2/a^2/e1^2 ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "K3:=op(1,H3/64/e2^2/a ^2/e1^2):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "factor(coeff(K2,e2 ,4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&,&I\"aG6\"\"\"\"F&F&\"\"#,(*& I#b2GF%F&I#b4GF%F&F&*&F*F&I#b5GF%F&!\"\"I\"BGF%F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "B:=solve(b2*b4-b2*b5+B,B);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&I#b2G6\"\"\"\"I#b4GF%F&!\"\"*&F$F&I#b5GF%F&F&" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "factor(coeff(K3,e2,4));" } {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*(,&I\"aG6\"\"\"\" F'F'\"\"#,&I#b4GF&F'I#b5GF&!\"\"F',&I#b2GF&F'I#b3GF&F'F'F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "b2:=-b3:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 23 "factor(coeff(K2,e2,2));" }{MPLTEXT 1 0 0 "" }} {PARA 11 "" 1 "" {XPPMATH 20 ",$**I#b3G6\"\"\"\"I#e1GF%\"\"#,&I#b4GF%F &I#b5GF%!\"\"F&,**$I\"aGF%F(F&*&F$F&F*F&F,*&F*F&F+F&F,F,F&F&!\"%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "factor(coeff(K3,e2,2));" } {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$**I#b3G6\"\"\"\"I# e1GF%\"\"#,&I#b4GF%F&I#b5GF%!\"\"F&,**$I\"aGF%F(F&*&F$F&F*F&F&*&F*F&F+ F&F,F,F&F&!\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "b4:=0:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "factor(coeff(K2,e2,2));" } {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*,I#b3G6\"\"\"\"I# b5GF%F&I#e1GF%\"\"#,&I\"aGF%F&!\"\"F&F&,&F+F&F&F&F&\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "a:=1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "factor(K2);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*.I#b3 G6\"\"\"\"I#b5GF%F&I#e1GF%F&I#e2GF%\"\"$,&F'F&F&F&F&,&F$F&F&F&F&\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "factor(K3);" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*.I#b3G6\"\"\"\"I#b5GF%F&I# e1GF%F&I#e2GF%\"\"$,&F'F&!\"\"F&F&,&F$F&F&F&F&\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "a:='a':" }{MPLTEXT 1 0 9 "b4:='b4':" } {MPLTEXT 1 0 16 "b2:='b2':B:='B':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "f1:='f1':" }{MPLTEXT 1 0 9 "f2:='f2':" }{MPLTEXT 1 0 9 "f3:='f3':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "# (b) T" }{MPLTEXT 1 0 42 "riples m1 ,m4,m5 and M1,M4,M5 are congruent" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "a:=1:b4:=0:b5:=B:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "solve(\{Phi,Delta21,Delta51\},\{f1,f2,f3\}):" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "assign(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "G4:=numer(simplify(Delta41)):nops(% );degree(G3,\{e0,e1,e2,e3\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#9" } }{PARA 11 "" 1 "" {XPPMATH 20 "\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "H4:=factor(resultant(G4,N,e0));" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$**I\"BG6\"\"\"#I#e1GF%F&I#e2GF%F&,&*&F$\"\"\"F'F+F+F(! \"\"F&\"#;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "a:='a':b4:='b 4':" }{MPLTEXT 1 0 9 "b5:='b5':" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "f1:='f1':f2:='f2':f3:='f3':e3:='e3':" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "# Item 2" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 6 "e2:=0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "a:=1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "# (a) T" }{MPLTEXT 1 0 57 "riples m1 ,m4,m5 and M1,M4,M5 are not reflection-congruent" }{MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "solve(\{Phi,Delta41,Delta5 1\},\{f1,f2,f3\}):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "assig n(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "G2:=numer(simplify (Delta21)):nops(%);degree(G2,\{e0,e1,e2,e3\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"$M\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "G3:=numer(simplify(Delta31)):nops(% );degree(G3,\{e0,e1,e2,e3\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"$M\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "H2:=factor(resultant(G2,N,e1)):nops(%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "H3:=factor(resultant(G3,N,e1)):nops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "K2:=o p(1,H2/16/e0" }{MPLTEXT 1 0 1 "^" }{MPLTEXT 1 0 3 "2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "K3:=op(1,H3/16/e0^2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "factor(coeff(K2,e0,2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "*(I\"BG6\"\"\"\"I#b4GF$F%,&I#b2GF$F%I#b5GF$!\"\" F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "factor(coeff(K3,e0,2) );" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "*(I\"BG6\"\"\" \"I#b4GF$F%,&I#b3GF$F%I#b5GF$!\"\"F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "# (i)" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "b4:=0:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 23 "factor(coeff(K2,e3,2));" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&I#b2G6\"\"\"\"I#b5GF%F&F&I\"BGF%! \"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "B:=b2*b5:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "factor(coeff(K3,e3,2));" } {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*&I#b5G6\"\"\"\",& I#b2GF%F&I#b3GF%F&F&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "b2:=-b3:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "factor(K2);" } }{PARA 11 "" 1 "" {XPPMATH 20 ",$*.I#b3G6\"\"\"\"I#b5GF%F&I#e0GF%F&I#e 3GF%F&,&F'F&!\"\"F&F&,&F$F&F+F&F&F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "factor(K3);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*.I#b3 G6\"\"\"\"I#b5GF%F&I#e0GF%F&I#e3GF%F&,&F'F&F&F&F&,&F$F&!\"\"F&F&F," }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "b4:='b4':b2:='b2':B:='B':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "# (ii)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "b2:=b5:b3:=b5:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "facto r(coeff(K2,e3,2));" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*&I#b4G6\"\"\"\"I#b5GF%F&!\"\"*$F'\"\"#F&I\"BGF%F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "B:=solve(-b4*b5+b5^2-B,B);" }} {PARA 11 "" 1 "" {XPPMATH 20 ",&*&I#b4G6\"\"\"\"I#b5GF%F&!\"\"*$F'\"\" #F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "factor(coeff(K3,e3,2 ));" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*&I#b5G6\"\" \"\",&I#b4GF%F&F$!\"\"F&\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "b3:='b3':b2:='b2':B:='B':" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "f1:='f1':f2:='f2':f3:='f3':e3:='e3':" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "# (b) " }{MPLTEXT 1 0 1 "T" } {MPLTEXT 1 0 53 "riples m1,m4,m5 and M1,M4,M5 are reflection-congruent " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "b4:=0:b5:=-B:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "solve(\{Phi,Delta21,Delta51 \},\{f1,f2,f3\}):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "assign (%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "G4:=numer(simplify( Delta41)):nops(%);degree(G3,\{e0,e1,e2,e3\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "H4:=factor(resultant(G4,N,e0));" }} {PARA 11 "" 1 "" {XPPMATH 20 ",$**I\"BG6\"\"\"#I#e3GF%F&,&*$I#e1GF%F& \"\"\"*$F'F&F+F+,(*&F$F&F*F&F+*&F$F&F'F&F+F,F+F+\"#;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "# " }{MPLTEXT 1 0 49 "This finishes the di scussion of the special case." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }