solution a8=1/2=b8 => a4=a5=a6=a7=b4=b5=b6=b7=0

Restent trois variables (a1,a2,a3) avec la règle

Length[{b1->-a1,b2->-a2,b3->-a3,a4->0,b4->0,a5->0,b5->0,a6->0,b6->0,a7->0,b7->0,a8->1/2,b8->1/2}]

N' entre pas dans le cas général :

eqd7a6/.{b1->-a1,b2->-a2,b3->-a3,a4->0,b4->0,a5->0,b5->0,a6->0,b6->0,a7->0,b7->0,a8->1/2,b8->1/2}

eq171na/.{b1->-a1,b2->-a2,b3->-a3,a4->0,b4->0,a5->0,b5->0,a6->0,b6->0,a7->0,b7->0,a8->1/2,b8->1/2}

L’équation est :

eq2/.a8->1/2

Très bien traitée dans le fichier, rien à ajouter .

solution a8=1/2 et b8=-1 => a4=a5=a6=a7=b4=b5=b6=b7=0

Restent trois variables (a1,a2,a3) avec la règle

Length[{b1->0,b2->0,b3->0,a4->0,b4->0,a5->0,b5->0,a6->0,b6->0,a7->0,b7->0,a8->1/2,b8->-1}]

N' entre pas dans le cas général :

eqd7a/.{b1->0,b2->0,b3->0,a4->0,b4->0,a5->0,b5->0,a6->0,b6->0,a7->0,b7->0,a8->1/2,b8->-1}

eqd7a6/.{b1->0,b2->0,b3->0,a4->0,b4->0,a5->0,b5->0,a6->0,b6->0,a7->0,b7->0,a8->1/2,b8->-1}

eq171na/.{b1->0,b2->0,b3->0,a4->0,b4->0,a5->0,b5->0,a6->0,b6->0,a7->0,b7->0,a8->1/2,b8->-1}

eq171nb/.{b1->0,b2->0,b3->0,a4->0,b4->0,a5->0,b5->0,a6->0,b6->0,a7->0,b7->0,a8->1/2,b8->-1}

Très bien traitée dans le fichier, rien à ajouter .

On peut noter que a4 = a5 = a6 = a7 = 0 donne a8 = 1/2 ou (a8= –1 et a1=a2=a3=0). Réciproquement,

eq1a/.a8->-1//Simplify

eq1a/.a8->1/2//Simplify

solution b8=1/2 et a8≠–1, a8≠1/2 => b4=b5=b6=b7=0. Identique à a8=1/2 et b8≠–1, b8≠1/2 => a4=a5=a6=a7=0 en inversant a↔b

Restent cinq variables (a4,a5,a6,a7,a8) avec la règle (on note que bi=-(3/2)(ai/(1+a8)))

Length[{a1->-(((Sqrt[3]) (a4 a6+a5 a7))/(-1+2 a8)),a2->-(((Sqrt[3]) (a5 a6-a4 a7))/(-1+2 a8)),a3->-(((Sqrt[3]) ((a4^2)+(a5^2)-(a6^2)-(a7^2)))/(2 (-1+2 a8))),b1->((3(Sqrt[3]) (a4 a6+a5 a7))/(2(-1+2 a8)(1+a8))),b2->((3(Sqrt[3]) (a5 a6-a4 a7))/(2(-1+2 a8)(1+a8))),b3->((3(Sqrt[3]) ((a4^2)+(a5^2)-(a6^2)-(a7^2)))/(4 (-1+2 a8)(1+a8))),a4->0,a5->0,a6->0,a7->0,a8->1/2}]

N' entre pas dans le cas général :

eqd7a/.{a1->-(((Sqrt[3]) (a4 a6+a5 a7))/(-1+2 a8)),a2->-(((Sqrt[3]) (a5 a6-a4 a7))/(-1+2 a8)),a3->-(((Sqrt[3]) ((a4^2)+(a5^2)-(a6^2)-(a7^2)))/(2 (-1+2 a8))),b1->((3(Sqrt[3]) (a4 a6+a5 a7))/(2(-1+2 a8)(1+a8))),b2->((3(Sqrt[3]) (a5 a6-a4 a7))/(2(-1+2 a8)(1+a8))),b3->((3(Sqrt[3]) ((a4^2)+(a5^2)-(a6^2)-(a7^2)))/(4 (-1+2 a8)(1+a8))),a4->0,a5->0,a6->0,a7->0,a8->1/2}

eqd7a6/.{a1->-(((Sqrt[3]) (a4 a6+a5 a7))/(-1+2 a8)),a2->-(((Sqrt[3]) (a5 a6-a4 a7))/(-1+2 a8)),a3->-(((Sqrt[3]) ((a4^2)+(a5^2)-(a6^2)-(a7^2)))/(2 (-1+2 a8))),b1->((3(Sqrt[3]) (a4 a6+a5 a7))/(2(-1+2 a8)(1+a8))),b2->((3(Sqrt[3]) (a5 a6-a4 a7))/(2(-1+2 a8)(1+a8))),b3->((3(Sqrt[3]) ((a4^2)+(a5^2)-(a6^2)-(a7^2)))/(4 (-1+2 a8)(1+a8))),a4->0,a5->0,a6->0,a7->0,a8->1/2}

eq171na/.{a1->-(((Sqrt[3]) (a4 a6+a5 a7))/(-1+2 a8)),a2->-(((Sqrt[3]) (a5 a6-a4 a7))/(-1+2 a8)),a3->-(((Sqrt[3]) ((a4^2)+(a5^2)-(a6^2)-(a7^2)))/(2 (-1+2 a8))),b1->((3(Sqrt[3]) (a4 a6+a5 a7))/(2(-1+2 a8)(1+a8))),b2->((3(Sqrt[3]) (a5 a6-a4 a7))/(2(-1+2 a8)(1+a8))),b3->((3(Sqrt[3]) ((a4^2)+(a5^2)-(a6^2)-(a7^2)))/(4 (-1+2 a8)(1+a8))),a4->0,a5->0,a6->0,a7->0,a8->1/2}

eq171nb/.{a1->-(((Sqrt[3]) (a4 a6+a5 a7))/(-1+2 a8)),a2->-(((Sqrt[3]) (a5 a6-a4 a7))/(-1+2 a8)),a3->-(((Sqrt[3]) ((a4^2)+(a5^2)-(a6^2)-(a7^2)))/(2 (-1+2 a8))),b1->((3(Sqrt[3]) (a4 a6+a5 a7))/(2(-1+2 a8)(1+a8))),b2->((3(Sqrt[3]) (a5 a6-a4 a7))/(2(-1+2 a8)(1+a8))),b3->((3(Sqrt[3]) ((a4^2)+(a5^2)-(a6^2)-(a7^2)))/(4 (-1+2 a8)(1+a8))),a4->0,a5->0,a6->0,a7->0,a8->1/2}

sol478na/.{a1->-(((Sqrt[3]) (a4 a6+a5 a7))/(-1+2 a8)),a2->-(((Sqrt[3]) (a5 a6-a4 a7))/(-1+2 a8)),a3->-(((Sqrt[3]) ((a4^2)+(a5^2)-(a6^2)-(a7^2)))/(2 (-1+2 a8))),b1->((3(Sqrt[3]) (a4 a6+a5 a7))/(2(-1+2 a8)(1+a8))),b2->((3(Sqrt[3]) (a5 a6-a4 a7))/(2(-1+2 a8)(1+a8))),b3->((3(Sqrt[3]) ((a4^2)+(a5^2)-(a6^2)-(a7^2)))/(4 (-1+2 a8)(1+a8))),a4->0,a5->0,a6->0,a7->0,a8->1/2}

Très bien traitée dans le fichier, l’équation finale est eq1a=0. Dans le cas symétrique, c’est eq1b=0. Il faut noter qu’exceptionnellement a8 FAIT PARTIE DES VARIABLES.

solution a1=a2=b1=b2=a6=a7=b6=b7=0 avec en plus a3=b3=((Sqrt[3])/4)qui apparaît pour le cas b8=a8=-(1/4)

Restent deux variables (a4,a5) avec la règle

Length[{a1->0,b1->0,a2->2,b2->0,a3->((Sqrt[3])/4),b3->a3,b4->-a4,b5->-a5,a6->0,b6->0,a7->0,b7->0,a8->-(1/4),b8->-(1/4)}]

N' entre pas dans le cas général (inutile de vérifier, aucun reg..a8 n’est valable) mais est un sous-cas de la solution 7

L’équation est (a4^2)+(a5^2)=(3/4).

Très bien traitée dans le fichier, le cas b8=-(1/2)-a8 plus général, émarge de la solution 7 avec {a6->0,b6->0,a7->0,b7->0}.

solution a1=a2=b1=b2=a4=a5=b4=b5=0 avec en plus a3=b3=((Sqrt[3])/4)qui apparaît pour le cas b8=a8=-(1/4).

Restent deux variables (a4,a5) avec la règle

Length[{a1->0,b1->0,a2->0,b2->0,a3->-((Sqrt[3])/4),b3->-((Sqrt[3])/4),b6->-a6,b7->-a7,a4->0,b4->0,a5->0,b5->0,a8->-(1/4),b8->-(1/4)}]

N' entre pas dans le cas général (inutile de vérifier, aucun reg..a8 n’est valable) mais est un sous-cas de la solution 7

L’équation est (a6^2)+(a7^2)=(3/4).

Très bien traitée dans le fichier, le cas b8=-(1/2)-a8 plus général, qui n’a pas été traité, émarge de la solution 7 avec {a4->0,b4->0,a5->0,b5->0}.

solution a4 b5–a5 b4=a4 b7+a7 b4=a4 b6+a6 b4=0.

Restent cinq variables (a4,a5,a6,a7,b4) avec la règle

Length[{a1->-((4 ((a4^2)+(a5^2)) (a4 a6+a5 a7) ((a6^2)+(a7^2)) b4)/((Sqrt[3]) ((a4^2)+(a5^2)-(a6^2)-(a7^2)) (a4 ((a4^2)+(a5^2)+(a6^2)+(a7^2))+((a4^2)+(a5^2)-(a6^2)-(a7^2)) b4))),b1->-((b4^2)/(a4^2))a1,a2->((4 ((a4^2)+(a5^2)) (-a5 a6+a4 a7) ((a6^2)+(a7^2)) b4)/((Sqrt[3]) ((a4^2)+(a5^2)-(a6^2)-(a7^2)) (a4 ((a4^2)+(a5^2)+(a6^2)+(a7^2))+((a4^2)+(a5^2)-(a6^2)-(a7^2)) b4))),b2->-((b4^2)/(a4^2))a2,a3->-((2 ((a4^2)+(a5^2)) ((a6^2)+(a7^2)) b4)/((Sqrt[3]) (a4 ((a4^2)+(a5^2)+(a6^2)+(a7^2))+((a4^2)+(a5^2)-(a6^2)-(a7^2)) b4))),b3->-((b4^2)/(a4^2))a3,b5->(b4/a4)a5,b6->-(b4/a4)a6,b7->-(b4/a4)a7,a8->-(1/4)+((3 (-((((a6^2)+(a7^2)))^2) (a4-b4)+((((a4^2)+(a5^2)))^2) (a4+b4)))/(8 ((a4^2)+(a5^2)) ((a6^2)+(a7^2)) b4)),b8->(1/2)-((2 ((a4^2)+(a5^2)) ((a6^2)+(a7^2)) (b4^2))/(a4 ((a4^3)+a4 ((a5^2)+(a6^2)+(a7^2))+(a4^2) b4+((a5^2)-(a6^2)-(a7^2)) b4)))}]

((a4^4)eqd7a/.{b5->(b4/a4)a5,b6->-(b4/a4)a6,b7->-(b4/a4)a7})==(a6^4) (b4^2) solsp1//Simplify

L’ équation est noté solsp1 dans le texte et émarge de eqd7a. Par contre, le cas spécifique a6=0, N’EMARGE PAS DE eqd7a mais de sol478na.

Restent alors quatre variables (a4,a5,a7,b4) avec la règle

Length[{a1->(((Sqrt[3]) a5 (((a4^2)+(a5^2)) (a4+b4)-(a7^2) (a4-b4)))/(4 ((a4^2)+(a5^2)) a7 b4)),b1->((4 a5 ((a4^2)+(a5^2)) (a7^3) (b4^2))/((Sqrt[3]) a4 ((a4^2)+(a5^2)-(a7^2)) (((a4^2)+(a5^2)) (a4+b4)-(a7^2) (a4-b4)))),a2->-(((Sqrt[3]) a4 (((a4^2)+(a5^2)) (a4+b4)-(a7^2) (a4-b4)))/(4 ((a4^2)+(a5^2)) a7 b4)),b2->-((4 ((a4^2)+(a5^2)) (a7^3) (b4^2))/((Sqrt[3]) ((a4^2)+(a5^2)-(a7^2)) (((a4^2)+(a5^2)) (a4+b4)-(a7^2) (a4-b4)))),a3->(((Sqrt[3]) ((a4^2)+(a5^2)-(a7^2)) (((a4^2)+(a5^2)) (a4+b4)-(a7^2) (a4-b4)))/(8 ((a4^2)+(a5^2)) (a7^2) b4)),b3->-((2 ((a4^2)+(a5^2)) (a7^2) (b4^2))/((Sqrt[3]) a4 (((a4^2)+(a5^2)) (a4+b4)-(a7^2) (a4-b4)))),b5->b4 a5/a4,b6->0, a6->0,b7->-b4 a7/a4,a8->(1/2)-((2 ((a4^2)+(a5^2)) (a7^2) b4)/( (((a4^2)+(a5^2)) (a4+b4)-(a7^2) (a4-b4)))),b8->(1/2)+((3 ((a4^2)+(a5^2)-(a7^2)) (((a4^2)+(a5^2)) (a4+b4)-(a7^2) (a4-b4)))/(8 a4 ((a4^2)+(a5^2)) (a7^2)))}]

l' équation  est solsp1a6 (la réduction de solsp1 quand a6=0)

solsp1a6

((cc[{a1,a2,a3,a4,a5,a6,a7,a8},{b1,b2,b3,b4,b5,b6,b7,b8}]+{a1,a2,a3,a4,a5,a6,a7,a8}+{b1,b2,b3,b4,b5,b6,b7,b8})/.{a1->(((Sqrt[3]) a5 (((a4^2)+(a5^2)) (a4+b4)-(a7^2) (a4-b4)))/(4 ((a4^2)+(a5^2)) a7 b4)),b1->((4 a5 ((a4^2)+(a5^2)) (a7^3) (b4^2))/((Sqrt[3]) a4 ((a4^2)+(a5^2)-(a7^2)) (((a4^2)+(a5^2)) (a4+b4)-(a7^2) (a4-b4)))),a2->-(((Sqrt[3]) a4 (((a4^2)+(a5^2)) (a4+b4)-(a7^2) (a4-b4)))/(4 ((a4^2)+(a5^2)) a7 b4)),b2->-((4 ((a4^2)+(a5^2)) (a7^3) (b4^2))/((Sqrt[3]) ((a4^2)+(a5^2)-(a7^2)) (((a4^2)+(a5^2)) (a4+b4)-(a7^2) (a4-b4)))),a3->(((Sqrt[3]) ((a4^2)+(a5^2)-(a7^2)) (((a4^2)+(a5^2)) (a4+b4)-(a7^2) (a4-b4)))/(8 ((a4^2)+(a5^2)) (a7^2) b4)),b3->-((2 ((a4^2)+(a5^2)) (a7^2) (b4^2))/((Sqrt[3]) a4 (((a4^2)+(a5^2)) (a4+b4)-(a7^2) (a4-b4)))),b5->b4 a5/a4,b6->0, a6->0,b7->-b4 a7/a4,a8->(1/2)-((2 ((a4^2)+(a5^2)) (a7^2) b4)/( (((a4^2)+(a5^2)) (a4+b4)-(a7^2) (a4-b4)))),b8->(1/2)+((3 ((a4^2)+(a5^2)-(a7^2)) (((a4^2)+(a5^2)) (a4+b4)-(a7^2) (a4-b4)))/(8 a4 ((a4^2)+(a5^2)) (a7^2)))}//Factor)=={(a5 solsp1a6 (3 (a7^6) (((a4-b4))^2)-3 ((((a4^2)+(a5^2)))^2) (a7^2) (a4-3 b4) (a4+b4)+3 ((((a4^2)+(a5^2)))^3) (((a4+b4))^2)-((a4^2)+(a5^2)) (a7^4) (3 (a4^2)+6 a4 b4+(-9+16 (a4^2)+16 (a5^2)) (b4^2))))/(32 (Sqrt[3]) a4 ((((a4^2)+(a5^2)))^2) (a7^3) ((a4^2)+(a5^2)-(a7^2)) b4 ((((a4^3)+a4 ((a5^2)-(a7^2))+(a4^2) b4+((a5^2)+(a7^2)) b4))^2)),-((solsp1a6 (3 (a7^6) (((a4-b4))^2)-3 ((((a4^2)+(a5^2)))^2) (a7^2) (a4-3 b4) (a4+b4)+3 ((((a4^2)+(a5^2)))^3) (((a4+b4))^2)-((a4^2)+(a5^2)) (a7^4) (3 (a4^2)+6 a4 b4+(-9+16 (a4^2)+16 (a5^2)) (b4^2))))/(32 (Sqrt[3]) ((((a4^2)+(a5^2)))^2) (a7^3) ((a4^2)+(a5^2)-(a7^2)) b4 ((((a4^3)+a4 ((a5^2)-(a7^2))+(a4^2) b4+((a5^2)+(a7^2)) b4))^2))),(solsp1a6 (3 (a7^6) (((a4-b4))^2)-3 ((((a4^2)+(a5^2)))^2) (a7^2) (a4-3 b4) (a4+b4)+3 ((((a4^2)+(a5^2)))^3) (((a4+b4))^2)+((a4^2)+(a5^2)) (a7^4) (-3 (a4^2)-6 a4 b4+(9+16 (a4^2)+16 (a5^2)) (b4^2))))/(64 (Sqrt[3]) a4 ((((a4^2)+(a5^2)))^2) (a7^4) b4 ((((a4^3)+a4 ((a5^2)-(a7^2))+(a4^2) b4+((a5^2)+(a7^2)) b4))^2)),solsp1a6/(8 ((a4^2)+(a5^2)) ((a4^2)+(a5^2)-(a7^2)) ((a4^3)+a4 ((a5^2)-(a7^2))+(a4^2) b4+((a5^2)+(a7^2)) b4)),(a5 solsp1a6)/(8 a4 ((a4^2)+(a5^2)) ((a4^2)+(a5^2)-(a7^2)) ((a4^3)+a4 ((a5^2)-(a7^2))+(a4^2) b4+((a5^2)+(a7^2)) b4)),0,solsp1a6/(8 a4 a7 ((a4^2)+(a5^2)-(a7^2)) ((a4^3)+a4 ((a5^2)-(a7^2))+(a4^2) b4+((a5^2)+(a7^2)) b4)),((a4^2)+(a5^2)+(a7^2)) solsp1a6/(16 a4 ((a4^2)+(a5^2)) (a7^2) ((a4^2)+(a5^2)-(a7^2)) ((a4^3)+a4 ((a5^2)-(a7^2))+(a4^2) b4+((a5^2)+(a7^2)) b4))}//Simplify

et elle dérive de sol478na (où apparaissent les cas limites pathologiques en facteur) (NB remplacement par sol478nou)

(a4^5)(sol478nou/.{b5->b4 a5/a4,b6->0, a6->0,b7->-b4 a7/a4})==((a4^2)+(a5^2)-(a7^2)) (b4^3) solsp1a6//Simplify

(a4^5) (sol478na/.{b5->b4 a5/a4,b6->0, a6->0,b7->-b4 a7/a4})==2a7 (b4^4) ((a7^2)-(a4^2)-(a5^2))solsp1a6//Simplify

solution a4+b4=a5+b5=a6+b6=a7+b7=0.

Restent cinq variables (a4,a5,a6,a7,a8) avec la règle

Length[{a1->-(((Sqrt[3]) (a4 a6+a5 a7))/(-1+2 a8)),b1->(((Sqrt[3]) (a4 a6+a5 a7))/(2 (1+a8))),a2->(((Sqrt[3]) (-a5 a6+a4 a7))/(-1+2 a8)),b2->(((Sqrt[3]) (a5 a6-a4 a7))/(2 (1+a8))),a3->(((Sqrt[3]) (-(a4^2)-(a5^2)+(a6^2)+(a7^2)))/(-2+4 a8)),b3->(((Sqrt[3]) ((a4^2)+(a5^2)-(a6^2)-(a7^2)))/(4 (1+a8))),b4->-a4,b5->-a5,b6->-a6,b7->-a7,b8->-(1/2)-a8}]

N' entre pas dans le cas général (inutile de vérifier, toutes les reg..a8 disparaissent, comme les reg48(a8),..,reg78 (a8)

L’équation est eq1a.

reg18==reg28==reg38/.{b4->-a4,b5->-a5,b6->-a6,b7->-a7}//Simplify

cnu48/.{b4->-a4,b5->-a5,b6->-a6,b7->-a7}//Simplify

On retrouve des cas particuliers qui ont échappé aux solutions 4 et 5 en appliquant les règles indiquées, alors, reste trois variables, soit (a4,a5,a8) avec

Length[{a1->0,b1->0,a2->2,b2->0,a3->((1+a8)/(Sqrt[3])),b3->((1-2 a8)/(2 (Sqrt[3]))),b4->-a4,b5->-a5,a6->0,b6->0,a7->0,b7->0,b8->-(1/2)-a8}]

soit (a6,a7,a8) avec

Length[{a1->0,b1->0,a2->2,b2->0,a3->-((1+a8)/(Sqrt[3])),b3->-((1-2 a8)/(2 (Sqrt[3]))),b4->-a4,b5->-a5,a6->0,b6->0,a7->0,b7->0,b8->-(1/2)-a8}]

solution a4 b5–a5 b4+a7 b6–a6 b7=0.

Restent sept variables (a4,a5,a6,a7,b4,b5,b6) avec la règle

Length[{a1->((Sqrt[3]) (a4 a6+a5 a7) (a6 a7 (a5 b4-a4 b5+a7 b6)+(((a5 b4-a4 b5+a7 b6))^2)+(a6^2) (a4 b4+(b4^2)+a5 b5+(b5^2)+a6 b6+(b6^2))))/(2 (a6^2) (((a5 b4-a4 b5))^2)+(a6^2) ((a6^2)+(a7^2)) ((b4^2)+(b5^2))-2 a6 (a4 b4+a5 b5) (a5 a7 b4-a4 a7 b5+((a6^2)+(a7^2)) b6)+((a4^2)+(a5^2)) ((a6^2) (b6^2)+(((a5 b4-a4 b5+a7 b6))^2))),b1->-(((a5 b4 b5-a4 (b5^2)+a6 b4 b6+a7 b5 b6) ((a5^4) (b4^2)+(a4^4) (b5^2)+(a6^2) ((a6^2)+(a7^2)) ((b4^2)+(b5^2))+2 (a5^3) a7 b4 b6-2 a5 a6 ((a6^2)+(a7^2)) b5 b6-2 (a4^3) b5 (a5 b4+a7 b6)-2 a4 ((a5^3) b4 b5+a5 a6 (2 a6 b4 b5+a7 (b4-b5) (b4+b5))+a6 ((a6^2)+(a7^2)) b4 b6+(a5^2) a7 b5 b6)+(a5^2) (2 a6 b4 (a6 b4-a7 b5)+((a6^2)+(a7^2)) (b6^2))+(a4^2) (2 a6 b5 (a7 b4+a6 b5)+(a5^2) ((b4^2)+(b5^2))+2 a5 a7 b4 b6+((a6^2)+(a7^2)) (b6^2))))/((Sqrt[3]) (a4 a6 b4+a5 a7 b4+a5 a6 b5-a4 a7 b5+((a6^2)+(a7^2)) b6) ((a5^2) (b4^2)+(a4^2) (b5^2)+(a6^2) ((b4^2)+(b5^2))+a6 ((a6^2)+(a7^2)) b6+((a6^2)+(a7^2)) (b6^2)+a5 (a6 a7 b4+(a6^2) b5+2 a7 b4 b6)+a4 ((a6^2) b4-a6 a7 b5-2 b5 (a5 b4+a7 b6))))),a2->((Sqrt[3]) (a5 a6-a4 a7) (a6 a7 (a5 b4-a4 b5+a7 b6)+(((a5 b4-a4 b5+a7 b6))^2)+(a6^2) (a4 b4+(b4^2)+a5 b5+(b5^2)+a6 b6+(b6^2))))/(2 (a6^2) (((a5 b4-a4 b5))^2)+(a6^2) ((a6^2)+(a7^2)) ((b4^2)+(b5^2))-2 a6 (a4 b4+a5 b5) (a5 a7 b4-a4 a7 b5+((a6^2)+(a7^2)) b6)+((a4^2)+(a5^2)) ((a6^2) (b6^2)+(((a5 b4-a4 b5+a7 b6))^2))),b2->((a5 (b4^2)-a4 b4 b5+a7 b4 b6-a6 b5 b6) ((a5^4) (b4^2)+(a4^4) (b5^2)+(a6^2) ((a6^2)+(a7^2)) ((b4^2)+(b5^2))+2 (a5^3) a7 b4 b6-2 a5 a6 ((a6^2)+(a7^2)) b5 b6-2 (a4^3) b5 (a5 b4+a7 b6)-2 a4 ((a5^3) b4 b5+a5 a6 (2 a6 b4 b5+a7 (b4-b5) (b4+b5))+a6 ((a6^2)+(a7^2)) b4 b6+(a5^2) a7 b5 b6)+(a5^2) (2 a6 b4 (a6 b4-a7 b5)+((a6^2)+(a7^2)) (b6^2))+(a4^2) (2 a6 b5 (a7 b4+a6 b5)+(a5^2) ((b4^2)+(b5^2))+2 a5 a7 b4 b6+((a6^2)+(a7^2)) (b6^2))))/((Sqrt[3]) (a4 a6 b4+a5 a7 b4+a5 a6 b5-a4 a7 b5+((a6^2)+(a7^2)) b6) ((a5^2) (b4^2)+(a4^2) (b5^2)+(a6^2) ((b4^2)+(b5^2))+a6 ((a6^2)+(a7^2)) b6+((a6^2)+(a7^2)) (b6^2)+a5 (a6 a7 b4+(a6^2) b5+2 a7 b4 b6)+a4 ((a6^2) b4-a6 a7 b5-2 b5 (a5 b4+a7 b6)))),a3->((Sqrt[3]) ((a4^2)+(a5^2)-(a6^2)-(a7^2)) (a6 a7 (a5 b4-a4 b5+a7 b6)+(((a5 b4-a4 b5+a7 b6))^2)+(a6^2) (a4 b4+(b4^2)+a5 b5+(b5^2)+a6 b6+(b6^2))))/(2 (2 (a6^2) (((a5 b4-a4 b5))^2)+(a6^2) ((a6^2)+(a7^2)) ((b4^2)+(b5^2))-2 a6 (a4 b4+a5 b5) (a5 a7 b4-a4 a7 b5+((a6^2)+(a7^2)) b6)+((a4^2)+(a5^2)) ((a6^2) (b6^2)+(((a5 b4-a4 b5+a7 b6))^2)))),b3->(((a5^2) (b4^2)+(((a4 b5-a7 b6))^2)+2 a5 b4 (-a4 b5+a7 b6)-(a6^2) ((b4^2)+(b5^2)-(b6^2))) ((a5^4) (b4^2)+(a4^4) (b5^2)+(a6^2) ((a6^2)+(a7^2)) ((b4^2)+(b5^2))+2 (a5^3) a7 b4 b6-2 a5 a6 ((a6^2)+(a7^2)) b5 b6-2 (a4^3) b5 (a5 b4+a7 b6)-2 a4 ((a5^3) b4 b5+a5 a6 (2 a6 b4 b5+a7 (b4-b5) (b4+b5))+a6 ((a6^2)+(a7^2)) b4 b6+(a5^2) a7 b5 b6)+(a5^2) (2 a6 b4 (a6 b4-a7 b5)+((a6^2)+(a7^2)) (b6^2))+(a4^2) (2 a6 b5 (a7 b4+a6 b5)+(a5^2) ((b4^2)+(b5^2))+2 a5 a7 b4 b6+((a6^2)+(a7^2)) (b6^2))))/(2 (Sqrt[3]) a6 (a4 a6 b4+a5 a7 b4+a5 a6 b5-a4 a7 b5+((a6^2)+(a7^2)) b6) ((a5^2) (b4^2)+(a4^2) (b5^2)+(a6^2) ((b4^2)+(b5^2))+a6 ((a6^2)+(a7^2)) b6+((a6^2)+(a7^2)) (b6^2)+a5 (a6 a7 b4+(a6^2) b5+2 a7 b4 b6)+a4 ((a6^2) b4-a6 a7 b5-2 b5 (a5 b4+a7 b6)))),b7->((a5 b4-a4 b5+a7 b6)/a6),a8->(1/2)-(2 (a6^2) (((a5 b4-a4 b5))^2)+(a6^2) ((a6^2)+(a7^2)) ((b4^2)+(b5^2))-2 a6 (a4 b4+a5 b5) (a7 (a5 b4-a4 b5)+((a6^2)+(a7^2)) b6)+((a4^2)+(a5^2)) ((a6^2) (b6^2)+(((a5 b4-a4 b5+a7 b6))^2)))/(2 (a6 a7 (a5 b4-a4 b5+a7 b6)+(((a5 b4-a4 b5+a7 b6))^2)+(a6^2) (a4 b4+(b4^2)+a5 b5+(b5^2)+a6 b6+(b6^2)))),b8->(1/2)+(3 (a7 (a5 b4-a4 b5)+a6 (a4 b4+a5 b5)+((a6^2)+(a7^2)) b6) (a6 a7 (a5 b4-a4 b5+a7 b6)+(((a5 b4-a4 b5+a7 b6))^2)+(a6^2) (b4 (a4+b4)+b5 (a5+b5)+b6 (a6+b6))))/(2 a6 (2 (a6^2) (((a5 b4-a4 b5))^2)+(a6^2) ((a6^2)+(a7^2)) ((b4^2)+(b5^2))-2 a6 (a4 b4+a5 b5) (a7 (a5 b4-a4 b5)+((a6^2)+(a7^2)) b6)+((a4^2)+(a5^2)) ((a6^2) (b6^2)+(((a5 b4-a4 b5+a7 b6))^2))))}]

L' équation eqd7a s' écrit

eqd7a

((cc[{a1,a2,a3,a4,a5,a6,a7,a8},{b1,b2,b3,b4,b5,b6,b7,b8}]+{a1,a2,a3,a4,a5,a6,a7,a8}+{b1,b2,b3,b4,b5,b6,b7,b8})/.{a1->((Sqrt[3]) (a4 a6+a5 a7) (a6 a7 (a5 b4-a4 b5+a7 b6)+(((a5 b4-a4 b5+a7 b6))^2)+(a6^2) (a4 b4+(b4^2)+a5 b5+(b5^2)+a6 b6+(b6^2))))/(2 (a6^2) (((a5 b4-a4 b5))^2)+(a6^2) ((a6^2)+(a7^2)) ((b4^2)+(b5^2))-2 a6 (a4 b4+a5 b5) (a5 a7 b4-a4 a7 b5+((a6^2)+(a7^2)) b6)+((a4^2)+(a5^2)) ((a6^2) (b6^2)+(((a5 b4-a4 b5+a7 b6))^2))),b1->-(((a5 b4 b5-a4 (b5^2)+a6 b4 b6+a7 b5 b6) ((a5^4) (b4^2)+(a4^4) (b5^2)+(a6^2) ((a6^2)+(a7^2)) ((b4^2)+(b5^2))+2 (a5^3) a7 b4 b6-2 a5 a6 ((a6^2)+(a7^2)) b5 b6-2 (a4^3) b5 (a5 b4+a7 b6)-2 a4 ((a5^3) b4 b5+a5 a6 (2 a6 b4 b5+a7 (b4-b5) (b4+b5))+a6 ((a6^2)+(a7^2)) b4 b6+(a5^2) a7 b5 b6)+(a5^2) (2 a6 b4 (a6 b4-a7 b5)+((a6^2)+(a7^2)) (b6^2))+(a4^2) (2 a6 b5 (a7 b4+a6 b5)+(a5^2) ((b4^2)+(b5^2))+2 a5 a7 b4 b6+((a6^2)+(a7^2)) (b6^2))))/((Sqrt[3]) (a4 a6 b4+a5 a7 b4+a5 a6 b5-a4 a7 b5+((a6^2)+(a7^2)) b6) ((a5^2) (b4^2)+(a4^2) (b5^2)+(a6^2) ((b4^2)+(b5^2))+a6 ((a6^2)+(a7^2)) b6+((a6^2)+(a7^2)) (b6^2)+a5 (a6 a7 b4+(a6^2) b5+2 a7 b4 b6)+a4 ((a6^2) b4-a6 a7 b5-2 b5 (a5 b4+a7 b6))))),a2->((Sqrt[3]) (a5 a6-a4 a7) (a6 a7 (a5 b4-a4 b5+a7 b6)+(((a5 b4-a4 b5+a7 b6))^2)+(a6^2) (a4 b4+(b4^2)+a5 b5+(b5^2)+a6 b6+(b6^2))))/(2 (a6^2) (((a5 b4-a4 b5))^2)+(a6^2) ((a6^2)+(a7^2)) ((b4^2)+(b5^2))-2 a6 (a4 b4+a5 b5) (a5 a7 b4-a4 a7 b5+((a6^2)+(a7^2)) b6)+((a4^2)+(a5^2)) ((a6^2) (b6^2)+(((a5 b4-a4 b5+a7 b6))^2))),b2->((a5 (b4^2)-a4 b4 b5+a7 b4 b6-a6 b5 b6) ((a5^4) (b4^2)+(a4^4) (b5^2)+(a6^2) ((a6^2)+(a7^2)) ((b4^2)+(b5^2))+2 (a5^3) a7 b4 b6-2 a5 a6 ((a6^2)+(a7^2)) b5 b6-2 (a4^3) b5 (a5 b4+a7 b6)-2 a4 ((a5^3) b4 b5+a5 a6 (2 a6 b4 b5+a7 (b4-b5) (b4+b5))+a6 ((a6^2)+(a7^2)) b4 b6+(a5^2) a7 b5 b6)+(a5^2) (2 a6 b4 (a6 b4-a7 b5)+((a6^2)+(a7^2)) (b6^2))+(a4^2) (2 a6 b5 (a7 b4+a6 b5)+(a5^2) ((b4^2)+(b5^2))+2 a5 a7 b4 b6+((a6^2)+(a7^2)) (b6^2))))/((Sqrt[3]) (a4 a6 b4+a5 a7 b4+a5 a6 b5-a4 a7 b5+((a6^2)+(a7^2)) b6) ((a5^2) (b4^2)+(a4^2) (b5^2)+(a6^2) ((b4^2)+(b5^2))+a6 ((a6^2)+(a7^2)) b6+((a6^2)+(a7^2)) (b6^2)+a5 (a6 a7 b4+(a6^2) b5+2 a7 b4 b6)+a4 ((a6^2) b4-a6 a7 b5-2 b5 (a5 b4+a7 b6)))),a3->((Sqrt[3]) ((a4^2)+(a5^2)-(a6^2)-(a7^2)) (a6 a7 (a5 b4-a4 b5+a7 b6)+(((a5 b4-a4 b5+a7 b6))^2)+(a6^2) (a4 b4+(b4^2)+a5 b5+(b5^2)+a6 b6+(b6^2))))/(2 (2 (a6^2) (((a5 b4-a4 b5))^2)+(a6^2) ((a6^2)+(a7^2)) ((b4^2)+(b5^2))-2 a6 (a4 b4+a5 b5) (a5 a7 b4-a4 a7 b5+((a6^2)+(a7^2)) b6)+((a4^2)+(a5^2)) ((a6^2) (b6^2)+(((a5 b4-a4 b5+a7 b6))^2)))),b3->(((a5^2) (b4^2)+(((a4 b5-a7 b6))^2)+2 a5 b4 (-a4 b5+a7 b6)-(a6^2) ((b4^2)+(b5^2)-(b6^2))) ((a5^4) (b4^2)+(a4^4) (b5^2)+(a6^2) ((a6^2)+(a7^2)) ((b4^2)+(b5^2))+2 (a5^3) a7 b4 b6-2 a5 a6 ((a6^2)+(a7^2)) b5 b6-2 (a4^3) b5 (a5 b4+a7 b6)-2 a4 ((a5^3) b4 b5+a5 a6 (2 a6 b4 b5+a7 (b4-b5) (b4+b5))+a6 ((a6^2)+(a7^2)) b4 b6+(a5^2) a7 b5 b6)+(a5^2) (2 a6 b4 (a6 b4-a7 b5)+((a6^2)+(a7^2)) (b6^2))+(a4^2) (2 a6 b5 (a7 b4+a6 b5)+(a5^2) ((b4^2)+(b5^2))+2 a5 a7 b4 b6+((a6^2)+(a7^2)) (b6^2))))/(2 (Sqrt[3]) a6 (a4 a6 b4+a5 a7 b4+a5 a6 b5-a4 a7 b5+((a6^2)+(a7^2)) b6) ((a5^2) (b4^2)+(a4^2) (b5^2)+(a6^2) ((b4^2)+(b5^2))+a6 ((a6^2)+(a7^2)) b6+((a6^2)+(a7^2)) (b6^2)+a5 (a6 a7 b4+(a6^2) b5+2 a7 b4 b6)+a4 ((a6^2) b4-a6 a7 b5-2 b5 (a5 b4+a7 b6)))),b7->((a5 b4-a4 b5+a7 b6)/a6),a8->(1/2)-(2 (a6^2) (((a5 b4-a4 b5))^2)+(a6^2) ((a6^2)+(a7^2)) ((b4^2)+(b5^2))-2 a6 (a4 b4+a5 b5) (a7 (a5 b4-a4 b5)+((a6^2)+(a7^2)) b6)+((a4^2)+(a5^2)) ((a6^2) (b6^2)+(((a5 b4-a4 b5+a7 b6))^2)))/(2 (a6 a7 (a5 b4-a4 b5+a7 b6)+(((a5 b4-a4 b5+a7 b6))^2)+(a6^2) (a4 b4+(b4^2)+a5 b5+(b5^2)+a6 b6+(b6^2)))),b8->(1/2)+(3 (a7 (a5 b4-a4 b5)+a6 (a4 b4+a5 b5)+((a6^2)+(a7^2)) b6) (a6 a7 (a5 b4-a4 b5+a7 b6)+(((a5 b4-a4 b5+a7 b6))^2)+(a6^2) (b4 (a4+b4)+b5 (a5+b5)+b6 (a6+b6))))/(2 a6 (2 (a6^2) (((a5 b4-a4 b5))^2)+(a6^2) ((a6^2)+(a7^2)) ((b4^2)+(b5^2))-2 a6 (a4 b4+a5 b5) (a7 (a5 b4-a4 b5)+((a6^2)+(a7^2)) b6)+((a4^2)+(a5^2)) ((a6^2) (b6^2)+(((a5 b4-a4 b5+a7 b6))^2))))})//Factor//Simplify

{(eqd7a ((a5^9) a6 (b4^5) b5-(a4^9) a6 (b5^6)+(a5^8) a6 (b4^4) (a6 b4+5 a7 b5) b6+(a6^5) ((((a6^2)+(a7^2)))^2) (a6 b4+a7 b5) ((((b4^2)+(b5^2)))^2) b6+(a4^8) a6 (b5^4) (5 a5 b4 b5+a6 b4 b6+5 a7 b5 b6)+(a5^7) (b4^3) (3 a7 (b4^3)+10 a6 (a7^2) b5 (b6^2)+2 (a6^3) b5 (2 (b4^2)+(b6^2))+4 (a6^2) a7 b4 (-(b5^2)+(b6^2)))-(a4^7) a6 (b5^3) (-3 (b5^3)+2 (a5^2) (5 (b4^2) b5+(b5^3))+4 a5 b4 (a6 b4+5 a7 b5) b6+10 (a7^2) b5 (b6^2)+4 a6 a7 b4 ((b5^2)+(b6^2))+2 (a6^2) b5 (2 (b5^2)+(b6^2)))+2 (a5^6) (b4^2) (9 (a7^2) (b4^3) b6+(a6^4) b4 b6 (2 (b4^2)-2 (b5^2)+(b6^2))+(a6^3) a7 b5 b6 (4 (b4^2)+3 (b6^2))+(a6^2) a7 b4 (3 b4 b5-8 a7 (b5^2) b6+3 a7 (b6^3))+a6 (a7^2) (3 (b4^3)+5 a7 b5 (b6^3)))+(a5^5) b4 (45 (a7^3) (b4^3) (b6^2)+5 a6 (a7^3) b6 (6 (b4^3)+a7 b5 (b6^3))+(a6^5) b5 (6 (b4^4)+2 (b4^2) (b5^2)+(b6^4))+(a6^4) a7 b4 ((b5^2) (3-8 (b4^2)-16 (b6^2))+4 (b6^2) (2 (b4^2)+(b6^2)))+2 (a6^3) a7 (3 (b4^3) b6+a7 b5 ((b4^4)+3 (b4^2) (1+(b5^2))+3 (b6^4)))+(a6^2) a7 b4 (24 a7 b4 b5 b6+9 (b4^2) ((b4^2)+(b5^2)+(b6^2))+(a7^2) (3 (b4^2)-24 (b5^2) (b6^2)+4 (b6^4))))+(a5^4) (60 (a7^4) (b4^3) (b6^3)+a6 (a7^4) (b6^2) (60 (b4^3)+a7 b5 (b6^3))+(a6^5) a7 b5 b6 (2 (b4^4)+(b6^4)+2 (b4^2) (3+7 (b5^2)-4 (b6^2)))+2 (a6^3) (a7^2) (6 (b4^5)+3 a7 (b4^4) b5 b6+a7 b5 (b6^5)+a7 (b4^2) b5 b6 (9+9 (b5^2)-4 (b6^2))+6 (b4^3) ((b5^2)+3 (b6^2)))+(a6^6) b4 b6 (6 (b4^4)-4 (b5^2) (b6^2)+(b6^4)+(b4^2) (-6 (b5^2)+4 (b6^2)))+(a6^2) (a7^2) b4 b6 (36 a7 b4 b5 b6+36 (b4^2) ((b4^2)+(b5^2)+(b6^2))+(a7^2) (12 (b4^2)-16 (b5^2) (b6^2)+(b6^4)))+2 (a6^4) a7 b4 (6 (b4^3) b5+a7 (b4^4) b6+6 b4 b5 ((b5^2)+(b6^2))+a7 (b4^2) b6 (3-5 (b5^2)+2 (b6^2))+a7 b6 ((b6^4)+(b5^2) (3-10 (b6^2)))))+2 (a5^2) (15 a6 (a7^6) b4 (b6^4)+9 (a7^6) b4 (b6^5)+(a6^8) b4 b6 (2 (b4^4)-2 (b5^4)+(b4^2) (b6^2)+3 (b5^2) (b6^2))+3 (a6^2) (a7^4) (b6^3) (2 (a7^2) b4+a7 b5 b6+6 b4 ((b4^2)+(b5^2)+(b6^2)))+(a6^3) (a7^4) (b6^2) (18 (b4^3)+a7 (b4^2) b5 b6+3 a7 b5 (1+(b5^2)) b6+6 b4 (3 (b5^2)+5 (b6^2)))+(a6^5) (a7^2) (3 (b4^5)+6 a7 b5 (b6^3)+3 a7 (b5^3) b6 (1+2 (b6^2))+a7 (b4^2) b5 b6 (3+2 (b6^2))+6 (b4^3) ((b5^2)+3 (b6^2))+3 b4 ((b5^4)+6 (b5^2) (b6^2)+5 (b6^4)))+(a6^7) a7 b5 b6 ((b4^2) (3+(b6^2))+3 ((b6^2)+(b5^2) (1+(b6^2))))+(a6^4) (a7^2) b6 (6 a7 b5 b6 ((b4^2)+(b5^2)+(b6^2))+9 b4 ((((b4^2)+(b5^2)+(b6^2)))^2)+(a7^2) b4 (-4 (b5^4)+12 (b6^2)+3 (b5^2) (1+(b6^2))+(b4^2) (3-4 (b5^2)+(b6^2))))+(a6^6) a7 (3 (b4^4) b5+2 a7 (b4^5) b6+6 (b4^2) b5 ((b5^2)+(b6^2))+3 b5 ((((b5^2)+(b6^2)))^2)+a7 (b4^3) b6 (3-4 (b5^2)+2 (b6^2))+3 a7 b4 b6 (-2 (b5^4)+2 (b6^2)+(b5^2) (1+2 (b6^2)))))+a5 (6 a6 (a7^7) (b6^5)+3 (a7^7) (b6^6)+(a6^9) b4 b5 ((b4^2)+(b5^2)) ((b4^2)+(b5^2)-4 (b6^2))+6 (a6^3) (a7^5) (b6^3) (2 (b4^2)+2 (b5^2)+3 (b6^2))+(a6^8) a7 (b6^2) (3 (b5^2)-4 (b5^4)+(b4^2) (3-4 (b5^2))+3 (b6^2))+3 (a6^2) (a7^5) (b6^4) ((a7^2)+3 ((b4^2)+(b5^2)+(b6^2)))+2 (a6^7) a7 (a7 b4 b5 ((b4^2)+(b5^2)) ((b4^2)+(b5^2)-4 (b6^2))+3 b6 ((((b4^2)+(b5^2)+(b6^2)))^2))+(a6^4) (a7^3) (b6^2) (9 ((((b4^2)+(b5^2)+(b6^2)))^2)+(a7^2) (3 (b5^2)-4 (b5^4)+(b4^2) (3-4 (b5^2))+9 (b6^2)))+(a6^6) 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(b5^4))+8 (a7^2) (b5^2) (b6^2)+9 (b6^4)+(a7^2) (18 (b6^2)-8 (b6^4))))+(a6^5) (12 b4 b5 b6 ((b4^2)+(b6^2))+a7 ((b4^4) (-3+6 (b5^2)+6 (b6^2))+(b5^2) (-2 (b5^4)+9 (b6^2)+6 (b5^2) (b6^2)+10 (b6^4))+(b4^2) (4 (b5^4)+(b5^2) (9-36 (b6^2))-(b6^2) (9+10 (b6^2)))))+(a6^3) (a7^2) (36 b4 b5 b6 ((b4^2)-(b5^2)+2 (b6^2))+a7 (2 (b5^6)-6 (b4^4) ((b5^2)-(b6^2))+(b5^4) (-3+6 (b6^2))+(b5^2) (b6^2) (9+20 (b6^2))-(b4^2) (4 (b5^4)+9 (b5^2) (-1+4 (b6^2))+(b6^2) (9+20 (b6^2)))))))))/(4 (Sqrt[3]) a6 (a4 a6 b4+a5 a7 b4+a5 a6 b5-a4 a7 b5+(a6^2) b6+(a7^2) b6) ((((a5^2) (b4^2)+(a6^2) (b4^2)+(a4^2) (b5^2)+(a6^2) (b5^2)+(a6^3) b6+a6 (a7^2) b6+(a6^2) (b6^2)+(a7^2) (b6^2)+a5 (a6 a7 b4+(a6^2) b5+2 a7 b4 b6)+a4 ((a6^2) b4-a6 a7 b5-2 b5 (a5 b4+a7 b6))))^2) ((((a5^4) (b4^2)+(a4^4) (b5^2)+(a6^2) ((a6^2)+(a7^2)) ((b4^2)+(b5^2))+2 (a5^3) a7 b4 b6-2 a5 a6 ((a6^2)+(a7^2)) b5 b6-2 (a4^3) b5 (a5 b4+a7 b6)-2 a4 ((a5^3) b4 b5+a5 a6 (2 a6 b4 b5+a7 ((b4^2)-(b5^2)))+a6 ((a6^2)+(a7^2)) b4 b6+(a5^2) a7 b5 b6)+(a5^2) (-2 a6 a7 b4 b5+(a7^2) (b6^2)+(a6^2) (2 (b4^2)+(b6^2)))+(a4^2) (2 a6 a7 b4 b5+(a5^2) ((b4^2)+(b5^2))+2 a5 a7 b4 b6+(a7^2) (b6^2)+(a6^2) (2 (b5^2)+(b6^2)))))^2))),(((a4^3) (b5^2)-2 (a4^2) b5 (a5 b4+a7 b6)-a6 b4 (a5 (a7 b4+a6 b5)+((a6^2)+(a7^2)) b6)+a4 ((a5^2) (b4^2)+a6 a7 b4 b5+2 a5 a7 b4 b6+(a7^2) (b6^2)+(a6^2) ((b5^2)+(b6^2)))) eqd7a)/(4 a6 (a4 a6 b4+a5 a7 b4+a5 a6 b5-a4 a7 b5+(a6^2) b6+(a7^2) b6) ((a5^2) (b4^2)+(a6^2) (b4^2)+(a4^2) (b5^2)+(a6^2) (b5^2)+(a6^3) b6+a6 (a7^2) b6+(a6^2) (b6^2)+(a7^2) (b6^2)+a5 (a6 a7 b4+(a6^2) b5+2 a7 b4 b6)+a4 ((a6^2) b4-a6 a7 b5-2 b5 (a5 b4+a7 b6))) ((a5^4) (b4^2)+(a4^4) (b5^2)+(a6^2) ((a6^2)+(a7^2)) ((b4^2)+(b5^2))+2 (a5^3) a7 b4 b6-2 a5 a6 ((a6^2)+(a7^2)) b5 b6-2 (a4^3) b5 (a5 b4+a7 b6)-2 a4 ((a5^3) b4 b5+a5 a6 (2 a6 b4 b5+a7 ((b4^2)-(b5^2)))+a6 ((a6^2)+(a7^2)) b4 b6+(a5^2) a7 b5 b6)+(a5^2) (-2 a6 a7 b4 b5+(a7^2) (b6^2)+(a6^2) (2 (b4^2)+(b6^2)))+(a4^2) (2 a6 a7 b4 b5+(a5^2) ((b4^2)+(b5^2))+2 a5 a7 b4 b6+(a7^2) (b6^2)+(a6^2) (2 (b5^2)+(b6^2))))),(((a5^3) (b4^2)-a6 b5 (a4 (a6 b4-a7 b5)+((a6^2)+(a7^2)) b6)+(a5^2) (-2 a4 b4 b5+2 a7 b4 b6)+a5 (-a6 a7 b4 b5+(((a4 b5-a7 b6))^2)+(a6^2) ((b4^2)+(b6^2)))) eqd7a)/(4 a6 (a4 a6 b4+a5 a7 b4+a5 a6 b5-a4 a7 b5+(a6^2) b6+(a7^2) b6) ((a5^2) (b4^2)+(a6^2) (b4^2)+(a4^2) (b5^2)+(a6^2) (b5^2)+(a6^3) b6+a6 (a7^2) b6+(a6^2) (b6^2)+(a7^2) (b6^2)+a5 (a6 a7 b4+(a6^2) b5+2 a7 b4 b6)+a4 ((a6^2) b4-a6 a7 b5-2 b5 (a5 b4+a7 b6))) ((a5^4) (b4^2)+(a4^4) (b5^2)+(a6^2) ((a6^2)+(a7^2)) ((b4^2)+(b5^2))+2 (a5^3) a7 b4 b6-2 a5 a6 ((a6^2)+(a7^2)) b5 b6-2 (a4^3) b5 (a5 b4+a7 b6)-2 a4 ((a5^3) b4 b5+a5 a6 (2 a6 b4 b5+a7 ((b4^2)-(b5^2)))+a6 ((a6^2)+(a7^2)) b4 b6+(a5^2) a7 b5 b6)+(a5^2) (-2 a6 a7 b4 b5+(a7^2) (b6^2)+(a6^2) (2 (b4^2)+(b6^2)))+(a4^2) (2 a6 a7 b4 b5+(a5^2) ((b4^2)+(b5^2))+2 a5 a7 b4 b6+(a7^2) (b6^2)+(a6^2) (2 (b5^2)+(b6^2))))),(((a5^2) (b4^2)+(a6^2) ((b4^2)+(b5^2))-a4 a6 b4 b6+a4 b5 (a4 b5-a7 b6)+a5 (-2 a4 b4 b5+a7 b4 b6-a6 b5 b6)) eqd7a)/(4 (a4 a6 b4+a5 a7 b4+a5 a6 b5-a4 a7 b5+(a6^2) b6+(a7^2) b6) ((a5^2) (b4^2)+(a6^2) (b4^2)+(a4^2) (b5^2)+(a6^2) (b5^2)+(a6^3) b6+a6 (a7^2) b6+(a6^2) (b6^2)+(a7^2) (b6^2)+a5 (a6 a7 b4+(a6^2) b5+2 a7 b4 b6)+a4 ((a6^2) b4-a6 a7 b5-2 b5 (a5 b4+a7 b6))) ((a5^4) (b4^2)+(a4^4) (b5^2)+(a6^2) ((a6^2)+(a7^2)) ((b4^2)+(b5^2))+2 (a5^3) a7 b4 b6-2 a5 a6 ((a6^2)+(a7^2)) b5 b6-2 (a4^3) b5 (a5 b4+a7 b6)-2 a4 ((a5^3) b4 b5+a5 a6 (2 a6 b4 b5+a7 ((b4^2)-(b5^2)))+a6 ((a6^2)+(a7^2)) b4 b6+(a5^2) a7 b5 b6)+(a5^2) (-2 a6 a7 b4 b5+(a7^2) (b6^2)+(a6^2) (2 (b4^2)+(b6^2)))+(a4^2) (2 a6 a7 b4 b5+(a5^2) ((b4^2)+(b5^2))+2 a5 a7 b4 b6+(a7^2) (b6^2)+(a6^2) (2 (b5^2)+(b6^2))))),(((a4^2) b4 b5+a6 a7 ((b4^2)+(b5^2))-a5 a6 b4 b6-a5 b5 (a5 b4+a7 b6)+a4 (a5 (-(b4^2)+(b5^2))-a7 b4 b6+a6 b5 b6)) eqd7a)/(4 (a4 a6 b4+a5 a7 b4+a5 a6 b5-a4 a7 b5+(a6^2) b6+(a7^2) b6) ((a5^2) (b4^2)+(a6^2) (b4^2)+(a4^2) (b5^2)+(a6^2) (b5^2)+(a6^3) b6+a6 (a7^2) b6+(a6^2) (b6^2)+(a7^2) (b6^2)+a5 (a6 a7 b4+(a6^2) b5+2 a7 b4 b6)+a4 ((a6^2) b4-a6 a7 b5-2 b5 (a5 b4+a7 b6))) ((a5^4) (b4^2)+(a4^4) (b5^2)+(a6^2) ((a6^2)+(a7^2)) ((b4^2)+(b5^2))+2 (a5^3) a7 b4 b6-2 a5 a6 ((a6^2)+(a7^2)) b5 b6-2 (a4^3) b5 (a5 b4+a7 b6)-2 a4 ((a5^3) b4 b5+a5 a6 (2 a6 b4 b5+a7 ((b4^2)-(b5^2)))+a6 ((a6^2)+(a7^2)) b4 b6+(a5^2) a7 b5 b6)+(a5^2) (-2 a6 a7 b4 b5+(a7^2) (b6^2)+(a6^2) (2 (b4^2)+(b6^2)))+(a4^2) (2 a6 a7 b4 b5+(a5^2) ((b4^2)+(b5^2))+2 a5 a7 b4 b6+(a7^2) (b6^2)+(a6^2) (2 (b5^2)+(b6^2))))),(((a5^2) (b4^2)+(((a4 b5-a7 b6))^2)+a5 (-2 a4 b4 b5+2 a7 b4 b6)+(a6^2) ((b4^2)+(b5^2)+(b6^2))) eqd7a)/(4 a6 (a4 a6 b4+a5 a7 b4+a5 a6 b5-a4 a7 b5+(a6^2) b6+(a7^2) b6) ((a5^2) (b4^2)+(a6^2) (b4^2)+(a4^2) (b5^2)+(a6^2) (b5^2)+(a6^3) b6+a6 (a7^2) b6+(a6^2) (b6^2)+(a7^2) (b6^2)+a5 (a6 a7 b4+(a6^2) b5+2 a7 b4 b6)+a4 ((a6^2) b4-a6 a7 b5-2 b5 (a5 b4+a7 b6))) ((a5^4) (b4^2)+(a4^4) (b5^2)+(a6^2) ((a6^2)+(a7^2)) ((b4^2)+(b5^2))+2 (a5^3) a7 b4 b6-2 a5 a6 ((a6^2)+(a7^2)) b5 b6-2 (a4^3) b5 (a5 b4+a7 b6)-2 a4 ((a5^3) b4 b5+a5 a6 (2 a6 b4 b5+a7 ((b4^2)-(b5^2)))+a6 ((a6^2)+(a7^2)) b4 b6+(a5^2) a7 b5 b6)+(a5^2) (-2 a6 a7 b4 b5+(a7^2) (b6^2)+(a6^2) (2 (b4^2)+(b6^2)))+(a4^2) (2 a6 a7 b4 b5+(a5^2) ((b4^2)+(b5^2))+2 a5 a7 b4 b6+(a7^2) (b6^2)+(a6^2) (2 (b5^2)+(b6^2)))))}

(({a1,a2,a3,a4,a5,a6,a7,a8}.{b1,b2,b3,b4,b5,b6,b7,b8}+1/2)/.{a1->((Sqrt[3]) (a4 a6+a5 a7) (a6 a7 (a5 b4-a4 b5+a7 b6)+(((a5 b4-a4 b5+a7 b6))^2)+(a6^2) (a4 b4+(b4^2)+a5 b5+(b5^2)+a6 b6+(b6^2))))/(2 (a6^2) (((a5 b4-a4 b5))^2)+(a6^2) ((a6^2)+(a7^2)) ((b4^2)+(b5^2))-2 a6 (a4 b4+a5 b5) (a5 a7 b4-a4 a7 b5+((a6^2)+(a7^2)) b6)+((a4^2)+(a5^2)) ((a6^2) (b6^2)+(((a5 b4-a4 b5+a7 b6))^2))),b1->-(((a5 b4 b5-a4 (b5^2)+a6 b4 b6+a7 b5 b6) ((a5^4) (b4^2)+(a4^4) (b5^2)+(a6^2) ((a6^2)+(a7^2)) ((b4^2)+(b5^2))+2 (a5^3) a7 b4 b6-2 a5 a6 ((a6^2)+(a7^2)) b5 b6-2 (a4^3) b5 (a5 b4+a7 b6)-2 a4 ((a5^3) b4 b5+a5 a6 (2 a6 b4 b5+a7 (b4-b5) (b4+b5))+a6 ((a6^2)+(a7^2)) b4 b6+(a5^2) a7 b5 b6)+(a5^2) (2 a6 b4 (a6 b4-a7 b5)+((a6^2)+(a7^2)) (b6^2))+(a4^2) (2 a6 b5 (a7 b4+a6 b5)+(a5^2) ((b4^2)+(b5^2))+2 a5 a7 b4 b6+((a6^2)+(a7^2)) (b6^2))))/((Sqrt[3]) (a4 a6 b4+a5 a7 b4+a5 a6 b5-a4 a7 b5+((a6^2)+(a7^2)) b6) ((a5^2) (b4^2)+(a4^2) (b5^2)+(a6^2) ((b4^2)+(b5^2))+a6 ((a6^2)+(a7^2)) b6+((a6^2)+(a7^2)) (b6^2)+a5 (a6 a7 b4+(a6^2) b5+2 a7 b4 b6)+a4 ((a6^2) b4-a6 a7 b5-2 b5 (a5 b4+a7 b6))))),a2->((Sqrt[3]) (a5 a6-a4 a7) (a6 a7 (a5 b4-a4 b5+a7 b6)+(((a5 b4-a4 b5+a7 b6))^2)+(a6^2) (a4 b4+(b4^2)+a5 b5+(b5^2)+a6 b6+(b6^2))))/(2 (a6^2) (((a5 b4-a4 b5))^2)+(a6^2) ((a6^2)+(a7^2)) ((b4^2)+(b5^2))-2 a6 (a4 b4+a5 b5) (a5 a7 b4-a4 a7 b5+((a6^2)+(a7^2)) b6)+((a4^2)+(a5^2)) ((a6^2) (b6^2)+(((a5 b4-a4 b5+a7 b6))^2))),b2->((a5 (b4^2)-a4 b4 b5+a7 b4 b6-a6 b5 b6) ((a5^4) (b4^2)+(a4^4) (b5^2)+(a6^2) ((a6^2)+(a7^2)) ((b4^2)+(b5^2))+2 (a5^3) a7 b4 b6-2 a5 a6 ((a6^2)+(a7^2)) b5 b6-2 (a4^3) b5 (a5 b4+a7 b6)-2 a4 ((a5^3) b4 b5+a5 a6 (2 a6 b4 b5+a7 (b4-b5) (b4+b5))+a6 ((a6^2)+(a7^2)) b4 b6+(a5^2) a7 b5 b6)+(a5^2) (2 a6 b4 (a6 b4-a7 b5)+((a6^2)+(a7^2)) (b6^2))+(a4^2) (2 a6 b5 (a7 b4+a6 b5)+(a5^2) ((b4^2)+(b5^2))+2 a5 a7 b4 b6+((a6^2)+(a7^2)) (b6^2))))/((Sqrt[3]) (a4 a6 b4+a5 a7 b4+a5 a6 b5-a4 a7 b5+((a6^2)+(a7^2)) b6) ((a5^2) (b4^2)+(a4^2) (b5^2)+(a6^2) ((b4^2)+(b5^2))+a6 ((a6^2)+(a7^2)) b6+((a6^2)+(a7^2)) (b6^2)+a5 (a6 a7 b4+(a6^2) b5+2 a7 b4 b6)+a4 ((a6^2) b4-a6 a7 b5-2 b5 (a5 b4+a7 b6)))),a3->((Sqrt[3]) ((a4^2)+(a5^2)-(a6^2)-(a7^2)) (a6 a7 (a5 b4-a4 b5+a7 b6)+(((a5 b4-a4 b5+a7 b6))^2)+(a6^2) (a4 b4+(b4^2)+a5 b5+(b5^2)+a6 b6+(b6^2))))/(2 (2 (a6^2) (((a5 b4-a4 b5))^2)+(a6^2) ((a6^2)+(a7^2)) ((b4^2)+(b5^2))-2 a6 (a4 b4+a5 b5) (a5 a7 b4-a4 a7 b5+((a6^2)+(a7^2)) b6)+((a4^2)+(a5^2)) ((a6^2) (b6^2)+(((a5 b4-a4 b5+a7 b6))^2)))),b3->(((a5^2) (b4^2)+(((a4 b5-a7 b6))^2)+2 a5 b4 (-a4 b5+a7 b6)-(a6^2) ((b4^2)+(b5^2)-(b6^2))) ((a5^4) (b4^2)+(a4^4) (b5^2)+(a6^2) ((a6^2)+(a7^2)) ((b4^2)+(b5^2))+2 (a5^3) a7 b4 b6-2 a5 a6 ((a6^2)+(a7^2)) b5 b6-2 (a4^3) b5 (a5 b4+a7 b6)-2 a4 ((a5^3) b4 b5+a5 a6 (2 a6 b4 b5+a7 (b4-b5) (b4+b5))+a6 ((a6^2)+(a7^2)) b4 b6+(a5^2) a7 b5 b6)+(a5^2) (2 a6 b4 (a6 b4-a7 b5)+((a6^2)+(a7^2)) (b6^2))+(a4^2) (2 a6 b5 (a7 b4+a6 b5)+(a5^2) ((b4^2)+(b5^2))+2 a5 a7 b4 b6+((a6^2)+(a7^2)) (b6^2))))/(2 (Sqrt[3]) a6 (a4 a6 b4+a5 a7 b4+a5 a6 b5-a4 a7 b5+((a6^2)+(a7^2)) b6) ((a5^2) (b4^2)+(a4^2) (b5^2)+(a6^2) ((b4^2)+(b5^2))+a6 ((a6^2)+(a7^2)) b6+((a6^2)+(a7^2)) (b6^2)+a5 (a6 a7 b4+(a6^2) b5+2 a7 b4 b6)+a4 ((a6^2) b4-a6 a7 b5-2 b5 (a5 b4+a7 b6)))),b7->((a5 b4-a4 b5+a7 b6)/a6),a8->(1/2)-(2 (a6^2) (((a5 b4-a4 b5))^2)+(a6^2) ((a6^2)+(a7^2)) ((b4^2)+(b5^2))-2 a6 (a4 b4+a5 b5) (a7 (a5 b4-a4 b5)+((a6^2)+(a7^2)) b6)+((a4^2)+(a5^2)) ((a6^2) (b6^2)+(((a5 b4-a4 b5+a7 b6))^2)))/(2 (a6 a7 (a5 b4-a4 b5+a7 b6)+(((a5 b4-a4 b5+a7 b6))^2)+(a6^2) (a4 b4+(b4^2)+a5 b5+(b5^2)+a6 b6+(b6^2)))),b8->(1/2)+(3 (a7 (a5 b4-a4 b5)+a6 (a4 b4+a5 b5)+((a6^2)+(a7^2)) b6) (a6 a7 (a5 b4-a4 b5+a7 b6)+(((a5 b4-a4 b5+a7 b6))^2)+(a6^2) (b4 (a4+b4)+b5 (a5+b5)+b6 (a6+b6))))/(2 a6 (2 (a6^2) (((a5 b4-a4 b5))^2)+(a6^2) ((a6^2)+(a7^2)) ((b4^2)+(b5^2))-2 a6 (a4 b4+a5 b5) (a7 (a5 b4-a4 b5)+((a6^2)+(a7^2)) b6)+((a4^2)+(a5^2)) ((a6^2) (b6^2)+(((a5 b4-a4 b5+a7 b6))^2))))}//Factor)==(((a5^2) (b4^2)+(((a4 b5-a7 b6))^2)+a5 (-2 a4 b4 b5+2 a7 b4 b6)+(a6^2) ((b4^2)+(b5^2)+(b6^2))) eqd7a)/(4 a6 (a4 a6 b4+a5 a7 b4+a5 a6 b5-a4 a7 b5+(a6^2) b6+(a7^2) b6) ((a5^2) (b4^2)+(a6^2) (b4^2)+(a4^2) (b5^2)+(a6^2) (b5^2)+(a6^3) b6+a6 (a7^2) b6+(a6^2) (b6^2)+(a7^2) (b6^2)+a5 (a6 a7 b4+(a6^2) b5+2 a7 b4 b6)+a4 ((a6^2) b4-a6 a7 b5-2 b5 (a5 b4+a7 b6))) ((a5^4) (b4^2)+(a4^4) (b5^2)+(a6^2) ((a6^2)+(a7^2)) ((b4^2)+(b5^2))+2 (a5^3) a7 b4 b6-2 a5 a6 ((a6^2)+(a7^2)) b5 b6-2 (a4^3) b5 (a5 b4+a7 b6)-2 a4 ((a5^3) b4 b5+a5 a6 (2 a6 b4 b5+a7 ((b4^2)-(b5^2)))+a6 ((a6^2)+(a7^2)) b4 b6+(a5^2) a7 b5 b6)+(a5^2) (-2 a6 a7 b4 b5+(a7^2) (b6^2)+(a6^2) (2 (b4^2)+(b6^2)))+(a4^2) (2 a6 a7 b4 b5+(a5^2) ((b4^2)+(b5^2))+2 a5 a7 b4 b6+(a7^2) (b6^2)+(a6^2) (2 (b5^2)+(b6^2)))))//Simplify

(eq1a/.{a1->((Sqrt[3]) (a4 a6+a5 a7) (a6 a7 (a5 b4-a4 b5+a7 b6)+(((a5 b4-a4 b5+a7 b6))^2)+(a6^2) (a4 b4+(b4^2)+a5 b5+(b5^2)+a6 b6+(b6^2))))/(2 (a6^2) (((a5 b4-a4 b5))^2)+(a6^2) ((a6^2)+(a7^2)) ((b4^2)+(b5^2))-2 a6 (a4 b4+a5 b5) (a5 a7 b4-a4 a7 b5+((a6^2)+(a7^2)) b6)+((a4^2)+(a5^2)) ((a6^2) (b6^2)+(((a5 b4-a4 b5+a7 b6))^2))),b1->-(((a5 b4 b5-a4 (b5^2)+a6 b4 b6+a7 b5 b6) ((a5^4) (b4^2)+(a4^4) (b5^2)+(a6^2) ((a6^2)+(a7^2)) ((b4^2)+(b5^2))+2 (a5^3) a7 b4 b6-2 a5 a6 ((a6^2)+(a7^2)) b5 b6-2 (a4^3) b5 (a5 b4+a7 b6)-2 a4 ((a5^3) b4 b5+a5 a6 (2 a6 b4 b5+a7 (b4-b5) (b4+b5))+a6 ((a6^2)+(a7^2)) b4 b6+(a5^2) a7 b5 b6)+(a5^2) (2 a6 b4 (a6 b4-a7 b5)+((a6^2)+(a7^2)) (b6^2))+(a4^2) (2 a6 b5 (a7 b4+a6 b5)+(a5^2) ((b4^2)+(b5^2))+2 a5 a7 b4 b6+((a6^2)+(a7^2)) (b6^2))))/((Sqrt[3]) (a4 a6 b4+a5 a7 b4+a5 a6 b5-a4 a7 b5+((a6^2)+(a7^2)) b6) ((a5^2) (b4^2)+(a4^2) (b5^2)+(a6^2) ((b4^2)+(b5^2))+a6 ((a6^2)+(a7^2)) b6+((a6^2)+(a7^2)) (b6^2)+a5 (a6 a7 b4+(a6^2) b5+2 a7 b4 b6)+a4 ((a6^2) b4-a6 a7 b5-2 b5 (a5 b4+a7 b6))))),a2->((Sqrt[3]) (a5 a6-a4 a7) (a6 a7 (a5 b4-a4 b5+a7 b6)+(((a5 b4-a4 b5+a7 b6))^2)+(a6^2) (a4 b4+(b4^2)+a5 b5+(b5^2)+a6 b6+(b6^2))))/(2 (a6^2) (((a5 b4-a4 b5))^2)+(a6^2) ((a6^2)+(a7^2)) ((b4^2)+(b5^2))-2 a6 (a4 b4+a5 b5) (a5 a7 b4-a4 a7 b5+((a6^2)+(a7^2)) b6)+((a4^2)+(a5^2)) ((a6^2) (b6^2)+(((a5 b4-a4 b5+a7 b6))^2))),b2->((a5 (b4^2)-a4 b4 b5+a7 b4 b6-a6 b5 b6) ((a5^4) (b4^2)+(a4^4) (b5^2)+(a6^2) ((a6^2)+(a7^2)) ((b4^2)+(b5^2))+2 (a5^3) a7 b4 b6-2 a5 a6 ((a6^2)+(a7^2)) b5 b6-2 (a4^3) b5 (a5 b4+a7 b6)-2 a4 ((a5^3) b4 b5+a5 a6 (2 a6 b4 b5+a7 (b4-b5) (b4+b5))+a6 ((a6^2)+(a7^2)) b4 b6+(a5^2) a7 b5 b6)+(a5^2) (2 a6 b4 (a6 b4-a7 b5)+((a6^2)+(a7^2)) (b6^2))+(a4^2) (2 a6 b5 (a7 b4+a6 b5)+(a5^2) ((b4^2)+(b5^2))+2 a5 a7 b4 b6+((a6^2)+(a7^2)) (b6^2))))/((Sqrt[3]) (a4 a6 b4+a5 a7 b4+a5 a6 b5-a4 a7 b5+((a6^2)+(a7^2)) b6) ((a5^2) (b4^2)+(a4^2) (b5^2)+(a6^2) ((b4^2)+(b5^2))+a6 ((a6^2)+(a7^2)) b6+((a6^2)+(a7^2)) (b6^2)+a5 (a6 a7 b4+(a6^2) b5+2 a7 b4 b6)+a4 ((a6^2) b4-a6 a7 b5-2 b5 (a5 b4+a7 b6)))),a3->((Sqrt[3]) ((a4^2)+(a5^2)-(a6^2)-(a7^2)) (a6 a7 (a5 b4-a4 b5+a7 b6)+(((a5 b4-a4 b5+a7 b6))^2)+(a6^2) (a4 b4+(b4^2)+a5 b5+(b5^2)+a6 b6+(b6^2))))/(2 (2 (a6^2) (((a5 b4-a4 b5))^2)+(a6^2) ((a6^2)+(a7^2)) ((b4^2)+(b5^2))-2 a6 (a4 b4+a5 b5) (a5 a7 b4-a4 a7 b5+((a6^2)+(a7^2)) b6)+((a4^2)+(a5^2)) ((a6^2) (b6^2)+(((a5 b4-a4 b5+a7 b6))^2)))),b3->(((a5^2) (b4^2)+(((a4 b5-a7 b6))^2)+2 a5 b4 (-a4 b5+a7 b6)-(a6^2) ((b4^2)+(b5^2)-(b6^2))) ((a5^4) (b4^2)+(a4^4) (b5^2)+(a6^2) ((a6^2)+(a7^2)) ((b4^2)+(b5^2))+2 (a5^3) a7 b4 b6-2 a5 a6 ((a6^2)+(a7^2)) b5 b6-2 (a4^3) b5 (a5 b4+a7 b6)-2 a4 ((a5^3) b4 b5+a5 a6 (2 a6 b4 b5+a7 (b4-b5) (b4+b5))+a6 ((a6^2)+(a7^2)) b4 b6+(a5^2) a7 b5 b6)+(a5^2) (2 a6 b4 (a6 b4-a7 b5)+((a6^2)+(a7^2)) (b6^2))+(a4^2) (2 a6 b5 (a7 b4+a6 b5)+(a5^2) ((b4^2)+(b5^2))+2 a5 a7 b4 b6+((a6^2)+(a7^2)) (b6^2))))/(2 (Sqrt[3]) a6 (a4 a6 b4+a5 a7 b4+a5 a6 b5-a4 a7 b5+((a6^2)+(a7^2)) b6) ((a5^2) (b4^2)+(a4^2) (b5^2)+(a6^2) ((b4^2)+(b5^2))+a6 ((a6^2)+(a7^2)) b6+((a6^2)+(a7^2)) (b6^2)+a5 (a6 a7 b4+(a6^2) b5+2 a7 b4 b6)+a4 ((a6^2) b4-a6 a7 b5-2 b5 (a5 b4+a7 b6)))),b7->((a5 b4-a4 b5+a7 b6)/a6),a8->(1/2)-(2 (a6^2) (((a5 b4-a4 b5))^2)+(a6^2) ((a6^2)+(a7^2)) ((b4^2)+(b5^2))-2 a6 (a4 b4+a5 b5) (a7 (a5 b4-a4 b5)+((a6^2)+(a7^2)) b6)+((a4^2)+(a5^2)) ((a6^2) (b6^2)+(((a5 b4-a4 b5+a7 b6))^2)))/(2 (a6 a7 (a5 b4-a4 b5+a7 b6)+(((a5 b4-a4 b5+a7 b6))^2)+(a6^2) (a4 b4+(b4^2)+a5 b5+(b5^2)+a6 b6+(b6^2)))),b8->(1/2)+(3 (a7 (a5 b4-a4 b5)+a6 (a4 b4+a5 b5)+((a6^2)+(a7^2)) b6) (a6 a7 (a5 b4-a4 b5+a7 b6)+(((a5 b4-a4 b5+a7 b6))^2)+(a6^2) (b4 (a4+b4)+b5 (a5+b5)+b6 (a6+b6))))/(2 a6 (2 (a6^2) (((a5 b4-a4 b5))^2)+(a6^2) ((a6^2)+(a7^2)) ((b4^2)+(b5^2))-2 a6 (a4 b4+a5 b5) (a7 (a5 b4-a4 b5)+((a6^2)+(a7^2)) b6)+((a4^2)+(a5^2)) ((a6^2) (b6^2)+(((a5 b4-a4 b5+a7 b6))^2))))}//Factor)==eqd7a/(3 ((((a5^2) (b4^2)+(a6^2) (b4^2)+(a4^2) (b5^2)+(a6^2) (b5^2)+(a6^3) b6+a6 (a7^2) b6+(a6^2) (b6^2)+(a7^2) (b6^2)+a5 (a6 a7 b4+(a6^2) b5+2 a7 b4 b6)+a4 ((a6^2) b4-a6 a7 b5-2 b5 (a5 b4+a7 b6))))^2))//Simplify

(eq1b/.{a1->((Sqrt[3]) (a4 a6+a5 a7) (a6 a7 (a5 b4-a4 b5+a7 b6)+(((a5 b4-a4 b5+a7 b6))^2)+(a6^2) (a4 b4+(b4^2)+a5 b5+(b5^2)+a6 b6+(b6^2))))/(2 (a6^2) (((a5 b4-a4 b5))^2)+(a6^2) ((a6^2)+(a7^2)) ((b4^2)+(b5^2))-2 a6 (a4 b4+a5 b5) (a5 a7 b4-a4 a7 b5+((a6^2)+(a7^2)) b6)+((a4^2)+(a5^2)) ((a6^2) (b6^2)+(((a5 b4-a4 b5+a7 b6))^2))),b1->-(((a5 b4 b5-a4 (b5^2)+a6 b4 b6+a7 b5 b6) ((a5^4) (b4^2)+(a4^4) (b5^2)+(a6^2) ((a6^2)+(a7^2)) ((b4^2)+(b5^2))+2 (a5^3) a7 b4 b6-2 a5 a6 ((a6^2)+(a7^2)) b5 b6-2 (a4^3) b5 (a5 b4+a7 b6)-2 a4 ((a5^3) b4 b5+a5 a6 (2 a6 b4 b5+a7 (b4-b5) (b4+b5))+a6 ((a6^2)+(a7^2)) b4 b6+(a5^2) a7 b5 b6)+(a5^2) (2 a6 b4 (a6 b4-a7 b5)+((a6^2)+(a7^2)) (b6^2))+(a4^2) (2 a6 b5 (a7 b4+a6 b5)+(a5^2) ((b4^2)+(b5^2))+2 a5 a7 b4 b6+((a6^2)+(a7^2)) (b6^2))))/((Sqrt[3]) (a4 a6 b4+a5 a7 b4+a5 a6 b5-a4 a7 b5+((a6^2)+(a7^2)) b6) ((a5^2) (b4^2)+(a4^2) (b5^2)+(a6^2) ((b4^2)+(b5^2))+a6 ((a6^2)+(a7^2)) b6+((a6^2)+(a7^2)) (b6^2)+a5 (a6 a7 b4+(a6^2) b5+2 a7 b4 b6)+a4 ((a6^2) b4-a6 a7 b5-2 b5 (a5 b4+a7 b6))))),a2->((Sqrt[3]) (a5 a6-a4 a7) (a6 a7 (a5 b4-a4 b5+a7 b6)+(((a5 b4-a4 b5+a7 b6))^2)+(a6^2) (a4 b4+(b4^2)+a5 b5+(b5^2)+a6 b6+(b6^2))))/(2 (a6^2) (((a5 b4-a4 b5))^2)+(a6^2) ((a6^2)+(a7^2)) ((b4^2)+(b5^2))-2 a6 (a4 b4+a5 b5) (a5 a7 b4-a4 a7 b5+((a6^2)+(a7^2)) b6)+((a4^2)+(a5^2)) ((a6^2) (b6^2)+(((a5 b4-a4 b5+a7 b6))^2))),b2->((a5 (b4^2)-a4 b4 b5+a7 b4 b6-a6 b5 b6) ((a5^4) (b4^2)+(a4^4) (b5^2)+(a6^2) ((a6^2)+(a7^2)) ((b4^2)+(b5^2))+2 (a5^3) a7 b4 b6-2 a5 a6 ((a6^2)+(a7^2)) b5 b6-2 (a4^3) b5 (a5 b4+a7 b6)-2 a4 ((a5^3) b4 b5+a5 a6 (2 a6 b4 b5+a7 (b4-b5) (b4+b5))+a6 ((a6^2)+(a7^2)) b4 b6+(a5^2) a7 b5 b6)+(a5^2) (2 a6 b4 (a6 b4-a7 b5)+((a6^2)+(a7^2)) (b6^2))+(a4^2) (2 a6 b5 (a7 b4+a6 b5)+(a5^2) ((b4^2)+(b5^2))+2 a5 a7 b4 b6+((a6^2)+(a7^2)) (b6^2))))/((Sqrt[3]) (a4 a6 b4+a5 a7 b4+a5 a6 b5-a4 a7 b5+((a6^2)+(a7^2)) b6) ((a5^2) (b4^2)+(a4^2) (b5^2)+(a6^2) ((b4^2)+(b5^2))+a6 ((a6^2)+(a7^2)) b6+((a6^2)+(a7^2)) (b6^2)+a5 (a6 a7 b4+(a6^2) b5+2 a7 b4 b6)+a4 ((a6^2) b4-a6 a7 b5-2 b5 (a5 b4+a7 b6)))),a3->((Sqrt[3]) ((a4^2)+(a5^2)-(a6^2)-(a7^2)) (a6 a7 (a5 b4-a4 b5+a7 b6)+(((a5 b4-a4 b5+a7 b6))^2)+(a6^2) (a4 b4+(b4^2)+a5 b5+(b5^2)+a6 b6+(b6^2))))/(2 (2 (a6^2) (((a5 b4-a4 b5))^2)+(a6^2) ((a6^2)+(a7^2)) ((b4^2)+(b5^2))-2 a6 (a4 b4+a5 b5) (a5 a7 b4-a4 a7 b5+((a6^2)+(a7^2)) b6)+((a4^2)+(a5^2)) ((a6^2) (b6^2)+(((a5 b4-a4 b5+a7 b6))^2)))),b3->(((a5^2) (b4^2)+(((a4 b5-a7 b6))^2)+2 a5 b4 (-a4 b5+a7 b6)-(a6^2) ((b4^2)+(b5^2)-(b6^2))) ((a5^4) (b4^2)+(a4^4) (b5^2)+(a6^2) ((a6^2)+(a7^2)) ((b4^2)+(b5^2))+2 (a5^3) a7 b4 b6-2 a5 a6 ((a6^2)+(a7^2)) b5 b6-2 (a4^3) b5 (a5 b4+a7 b6)-2 a4 ((a5^3) b4 b5+a5 a6 (2 a6 b4 b5+a7 (b4-b5) (b4+b5))+a6 ((a6^2)+(a7^2)) b4 b6+(a5^2) a7 b5 b6)+(a5^2) (2 a6 b4 (a6 b4-a7 b5)+((a6^2)+(a7^2)) (b6^2))+(a4^2) (2 a6 b5 (a7 b4+a6 b5)+(a5^2) ((b4^2)+(b5^2))+2 a5 a7 b4 b6+((a6^2)+(a7^2)) (b6^2))))/(2 (Sqrt[3]) a6 (a4 a6 b4+a5 a7 b4+a5 a6 b5-a4 a7 b5+((a6^2)+(a7^2)) b6) ((a5^2) (b4^2)+(a4^2) (b5^2)+(a6^2) ((b4^2)+(b5^2))+a6 ((a6^2)+(a7^2)) b6+((a6^2)+(a7^2)) (b6^2)+a5 (a6 a7 b4+(a6^2) b5+2 a7 b4 b6)+a4 ((a6^2) b4-a6 a7 b5-2 b5 (a5 b4+a7 b6)))),b7->((a5 b4-a4 b5+a7 b6)/a6),a8->(1/2)-(2 (a6^2) (((a5 b4-a4 b5))^2)+(a6^2) ((a6^2)+(a7^2)) ((b4^2)+(b5^2))-2 a6 (a4 b4+a5 b5) (a7 (a5 b4-a4 b5)+((a6^2)+(a7^2)) b6)+((a4^2)+(a5^2)) ((a6^2) (b6^2)+(((a5 b4-a4 b5+a7 b6))^2)))/(2 (a6 a7 (a5 b4-a4 b5+a7 b6)+(((a5 b4-a4 b5+a7 b6))^2)+(a6^2) (a4 b4+(b4^2)+a5 b5+(b5^2)+a6 b6+(b6^2)))),b8->(1/2)+(3 (a7 (a5 b4-a4 b5)+a6 (a4 b4+a5 b5)+((a6^2)+(a7^2)) b6) (a6 a7 (a5 b4-a4 b5+a7 b6)+(((a5 b4-a4 b5+a7 b6))^2)+(a6^2) (b4 (a4+b4)+b5 (a5+b5)+b6 (a6+b6))))/(2 a6 (2 (a6^2) (((a5 b4-a4 b5))^2)+(a6^2) ((a6^2)+(a7^2)) ((b4^2)+(b5^2))-2 a6 (a4 b4+a5 b5) (a7 (a5 b4-a4 b5)+((a6^2)+(a7^2)) b6)+((a4^2)+(a5^2)) ((a6^2) (b6^2)+(((a5 b4-a4 b5+a7 b6))^2))))}//Factor)==(((a5^2) (b4^2)+(((a4 b5-a7 b6))^2)+a5 (-2 a4 b4 b5+2 a7 b4 b6)+(a6^2) ((b4^2)+(b5^2)+(b6^2))) eqd7a)/((a6^2) ((((a5^4) (b4^2)+(a4^4) (b5^2)+(a6^2) ((a6^2)+(a7^2)) ((b4^2)+(b5^2))+2 (a5^3) a7 b4 b6-2 a5 a6 ((a6^2)+(a7^2)) b5 b6-2 (a4^3) b5 (a5 b4+a7 b6)-2 a4 ((a5^3) b4 b5+a5 a6 (2 a6 b4 b5+a7 ((b4^2)-(b5^2)))+a6 ((a6^2)+(a7^2)) b4 b6+(a5^2) a7 b5 b6)+(a5^2) (-2 a6 a7 b4 b5+(a7^2) (b6^2)+(a6^2) (2 (b4^2)+(b6^2)))+(a4^2) (2 a6 a7 b4 b5+(a5^2) ((b4^2)+(b5^2))+2 a5 a7 b4 b6+(a7^2) (b6^2)+(a6^2) (2 (b5^2)+(b6^2)))))^2))//Simplify

Restent sept variables (a4,a5,a6,b4,b5,b6,b7) avec la règle symétrique et on trouve eqd7 où a↔b (noté eqd7a6)

solution a4=a5=a6=b6=0.

Restent quatre variables (b4,b5,a7,b7) avec la règle

Length[{a1->0,b1->-((a7 b5 ((b4^2)+(b5^2)))/((Sqrt[3]) ((b4^2)+(b5^2)+b7 (a7+b7)))),a2->0,b2->((a7 b4 ((b4^2)+(b5^2)))/((Sqrt[3]) ((b4^2)+(b5^2)+b7 (a7+b7)))),a3->(((a7^2) ((b4^2)+(b5^2)))/(2 (Sqrt[3]) b7 (a7+b7))),b3->-((a7 ((b4^2)+(b5^2)) ((b4^2)+(b5^2)-(b7^2)))/(2 (Sqrt[3]) b7 ((b4^2)+(b5^2)+b7 (a7+b7)))),a4->0,a5->0,a6->0,b6->0,a8->(((b4^2)+(b5^2)+3 b7 (a7+b7))/(2 ((b4^2)+(b5^2)))),b8->(1/2)+((3 b7 ((b4^2)+(b5^2)+b7 (a7+b7)))/(2 a7 ((b4^2)+(b5^2))))}]

L' équation est solspe168

(cc[{a1,a2,a3,a4,a5,a6,a7,a8},{b1,b2,b3,b4,b5,b6,b7,b8}]+{a1,a2,a3,a4,a5,a6,a7,a8}+{b1,b2,b3,b4,b5,b6,b7,b8}/.{a1->0,b1->-((a7 b5 ((b4^2)+(b5^2)))/((Sqrt[3]) ((b4^2)+(b5^2)+b7 (a7+b7)))),a2->0,b2->((a7 b4 ((b4^2)+(b5^2)))/((Sqrt[3]) ((b4^2)+(b5^2)+b7 (a7+b7)))),a3->(((a7^2) ((b4^2)+(b5^2)))/(2 (Sqrt[3]) b7 (a7+b7))),b3->-((a7 ((b4^2)+(b5^2)) ((b4^2)+(b5^2)-(b7^2)))/(2 (Sqrt[3]) b7 ((b4^2)+(b5^2)+b7 (a7+b7)))),a4->0,a5->0,a6->0,b6->0,a8->(((b4^2)+(b5^2)+3 b7 (a7+b7))/(2 ((b4^2)+(b5^2)))),b8->(1/2)+((3 b7 ((b4^2)+(b5^2)+b7 (a7+b7)))/(2 a7 ((b4^2)+(b5^2))))}//Factor)=={0,0,0,((b4 ((b4^2)+(b5^2)-b7 (a7+b7)) ((eq171na/((a7^5) (b5^2)))))/(4 ((b4^2)+(b5^2)) b7 (a7+b7) ((b4^2)+(b5^2)+b7 (a7+b7)))),((b5 ((b4^2)+(b5^2)-b7 (a7+b7)) ((eq171na/((a7^5) (b5^2)))))/(4 ((b4^2)+(b5^2)) b7 (a7+b7) ((b4^2)+(b5^2)+b7 (a7+b7)))),0,(((-2 (b7^3)+a7 ((b4^2)+(b5^2)-2 (b7^2))) ((eq171na/((a7^5) (b5^2)))))/(4 ((b4^2)+(b5^2)) b7 (a7+b7) ((b4^2)+(b5^2)+b7 (a7+b7)))),-((((eq171na/((a7^5) (b5^2)))) ((a7^2) ((((b4^2)+(b5^2)))^3)-((a7^2) (b7^2)+3 (a7+b7)b7)((((b4^2)+(b5^2)))^2)+3 ((b4^2)+(b5^2)) (b7^2) (-(a7^2)+a7 b7+2 (b7^2))+9 (b7^4) (((a7+b7))^2)))/(12 a7 ((((b4^2)+(b5^2)))^2) (b7^2) (a7+b7) ((b4^2)+(b5^2)+b7 (a7+b7))))}/.{a4->0,a5->0,a6->0,b6->0}//Simplify

({a1,a2,a3,a4,a5,a6,a7,a8}.{b1,b2,b3,b4,b5,b6,b7,b8}+1/2/.{a1->0,b1->-((a7 b5 ((b4^2)+(b5^2)))/((Sqrt[3]) ((b4^2)+(b5^2)+b7 (a7+b7)))),a2->0,b2->((a7 b4 ((b4^2)+(b5^2)))/((Sqrt[3]) ((b4^2)+(b5^2)+b7 (a7+b7)))),a3->(((a7^2) ((b4^2)+(b5^2)))/(2 (Sqrt[3]) b7 (a7+b7))),b3->-((a7 ((b4^2)+(b5^2)) ((b4^2)+(b5^2)-(b7^2)))/(2 (Sqrt[3]) b7 ((b4^2)+(b5^2)+b7 (a7+b7)))),a4->0,a5->0,a6->0,b6->0,a8->(((b4^2)+(b5^2)+3 b7 (a7+b7))/(2 ((b4^2)+(b5^2)))),b8->(1/2)+((3 b7 ((b4^2)+(b5^2)+b7 (a7+b7)))/(2 a7 ((b4^2)+(b5^2))))}//Factor)==-((eq171na/((a7^5) (b5^2)))) ((a7^2) ((((b4^2)+(b5^2)))^3)-((a7^2) (b7^2)+3 (a7+b7)b7)((((b4^2)+(b5^2)))^2)-3 ((b4^2)+(b5^2)) (b7^2) (a7+4 b7)(a7+b7)-9 (b7^4) (((a7+b7))^2))/(12 a7 ((((b4^2)+(b5^2)))^2) (b7^2) (a7+b7) ((b4^2)+(b5^2)+b7 (a7+b7)))/.{a4->0,a5->0,a6->0,b6->0}//Simplify

(eq1a/.{a1->0,b1->-((a7 b5 ((b4^2)+(b5^2)))/((Sqrt[3]) ((b4^2)+(b5^2)+b7 (a7+b7)))),a2->0,b2->((a7 b4 ((b4^2)+(b5^2)))/((Sqrt[3]) ((b4^2)+(b5^2)+b7 (a7+b7)))),a3->(((a7^2) ((b4^2)+(b5^2)))/(2 (Sqrt[3]) b7 (a7+b7))),b3->-((a7 ((b4^2)+(b5^2)) ((b4^2)+(b5^2)-(b7^2)))/(2 (Sqrt[3]) b7 ((b4^2)+(b5^2)+b7 (a7+b7)))),a4->0,a5->0,a6->0,b6->0,a8->(((b4^2)+(b5^2)+3 b7 (a7+b7))/(2 ((b4^2)+(b5^2)))),b8->(1/2)+((3 b7 ((b4^2)+(b5^2)+b7 (a7+b7)))/(2 a7 ((b4^2)+(b5^2))))}//Factor)==((eq171na/((a7^5) (b5^2)((((b4^2)+(b5^2)))^2))))/.{a4->0,a5->0,a6->0,b6->0}//Simplify

(eq1b/.{a1->0,b1->-((a7 b5 ((b4^2)+(b5^2)))/((Sqrt[3]) ((b4^2)+(b5^2)+b7 (a7+b7)))),a2->0,b2->((a7 b4 ((b4^2)+(b5^2)))/((Sqrt[3]) ((b4^2)+(b5^2)+b7 (a7+b7)))),a3->(((a7^2) ((b4^2)+(b5^2)))/(2 (Sqrt[3]) b7 (a7+b7))),b3->-((a7 ((b4^2)+(b5^2)) ((b4^2)+(b5^2)-(b7^2)))/(2 (Sqrt[3]) b7 ((b4^2)+(b5^2)+b7 (a7+b7)))),a4->0,a5->0,a6->0,b6->0,a8->(((b4^2)+(b5^2)+3 b7 (a7+b7))/(2 ((b4^2)+(b5^2)))),b8->(1/2)+((3 b7 ((b4^2)+(b5^2)+b7 (a7+b7)))/(2 a7 ((b4^2)+(b5^2))))}//Factor)==(((((b4^2)+(b5^2)+(b7^2))eq171na)/((a7^7) (b5^2)((((b4^2)+(b5^2)))^2))))/.{a4->0,a5->0,a6->0,b6->0}//Simplify

N' entre pas dans le cas eqd7a ni eqd7a6 :

eqd7a/.{a4->0,a5->0,a6->0,b6->0}//Simplify

eqd7a6/.{a4->0,a5->0,a6->0,b6->0}//Simplify

mais l’équation eq171na a été créé pour couvrir ce cas (eq171na est un des universels d’eqd7a et eq171nb d’eqd7a6). NB solspe168 fonctionne aussi ici.

(eq171na/.{a4->0,a5->0,a6->0,b6->0})==(a7^5) (b5^2)solspe168//Factor

(sol478nou/.{a4->0,a5->0,a6->0,b6->0})==(a7^3)b7 solspe168//Simplify

solution a4 b5–a5 b4=a6=b6=0 et a4≠0.

Restent cinq variables (a4,a5,b4,b5,b7) avec la règle

Length[{a1->(((Sqrt[3]) a5 a7 ((a4+b4) ((a4^2)+(a5^2)) b4+(a4^2) b7 (a7+b7)))/(((a4^2)+(a5^2)) (((a7 b4-a4 b7))^2))),b1->-((a5 ((a4^2)+(a5^2)) b4 b7 (((a7 b4-a4 b7))^2))/((Sqrt[3]) (((a4^2)+(a5^2)) b4+a4 a7 b7) (((a4^2)+(a5^2)) b4 (a4+b4)+(a4^2) (a7+b7) b7))),a2->-(((Sqrt[3]) a4 a7 ((a4+b4) ((a4^2)+(a5^2)) b4+(a4^2) b7 (a7+b7)))/(((a4^2)+(a5^2)) (((a7 b4-a4 b7))^2))),b2->((a4 ((a4^2)+(a5^2)) b4 b7 (((a7 b4-a4 b7))^2))/((Sqrt[3]) (((a4^2)+(a5^2)) b4+a4 a7 b7) (((a4^2)+(a5^2)) b4 (a4+b4)+(a4^2) (a7+b7) b7))),a3->(((Sqrt[3]) ((a4^2)+(a5^2)-(a7^2)) ((a4+b4) ((a4^2)+(a5^2)) b4+(a4^2) b7 (a7+b7)))/(2 ((a4^2)+(a5^2)) (((a7 b4-a4 b7))^2))),b3->-((((a4^2)+(a5^2)) (((a7 b4-a4 b7))^2) (((a4^2)+(a5^2)) (b4^2)-(a4^2)(b7^2)))/(2 (Sqrt[3]) a4(((a4^2)+(a5^2)) b4+a4 a7 b7) (((a4^2)+(a5^2)) b4 (a4+b4)+(a4^2) (a7+b7) b7))),b5->a5 b4/a4,a6->0,b6->0,a8->(1/2)-((((a4^2)+(a5^2)) (((a7 b4-a4 b7))^2))/(2 (((a4^2)+(a5^2)) b4 (a4+b4)+(a4^2) (a7+b7) b7))),b8->(1/2)+((3 (((a4^2)+(a5^2)) b4+a4 a7 b7) ((a4+b4) ((a4^2)+(a5^2)) b4+(a4^2) b7 (a7+b7)))/(2 a4 ((a4^2)+(a5^2)) (((a7 b4-a4 b7))^2)))}]

L’équation s’écrit

sol141nsp1b6

((cc[{a1,a2,a3,a4,a5,a6,a7,a8},{b1,b2,b3,b4,b5,b6,b7,b8}]+{a1,a2,a3,a4,a5,a6,a7,a8}+{b1,b2,b3,b4,b5,b6,b7,b8})/.{a1->(((Sqrt[3]) a5 a7 ((a4+b4) ((a4^2)+(a5^2)) b4+(a4^2) b7 (a7+b7)))/(((a4^2)+(a5^2)) (((a7 b4-a4 b7))^2))),b1->-((a5 ((a4^2)+(a5^2)) b4 b7 (((a7 b4-a4 b7))^2))/((Sqrt[3]) (((a4^2)+(a5^2)) b4+a4 a7 b7) (((a4^2)+(a5^2)) b4 (a4+b4)+(a4^2) (a7+b7) b7))),a2->-(((Sqrt[3]) a4 a7 ((a4+b4) ((a4^2)+(a5^2)) b4+(a4^2) b7 (a7+b7)))/(((a4^2)+(a5^2)) (((a7 b4-a4 b7))^2))),b2->((a4 ((a4^2)+(a5^2)) b4 b7 (((a7 b4-a4 b7))^2))/((Sqrt[3]) (((a4^2)+(a5^2)) b4+a4 a7 b7) (((a4^2)+(a5^2)) b4 (a4+b4)+(a4^2) (a7+b7) b7))),a3->(((Sqrt[3]) ((a4^2)+(a5^2)-(a7^2)) ((a4+b4) ((a4^2)+(a5^2)) b4+(a4^2) b7 (a7+b7)))/(2 ((a4^2)+(a5^2)) (((a7 b4-a4 b7))^2))),b3->-((((a4^2)+(a5^2)) (((a7 b4-a4 b7))^2) (((a4^2)+(a5^2)) (b4^2)-(a4^2)(b7^2)))/(2 (Sqrt[3]) a4(((a4^2)+(a5^2)) b4+a4 a7 b7) (((a4^2)+(a5^2)) b4 (a4+b4)+(a4^2) (a7+b7) b7))),b5->a5 b4/a4,a6->0,b6->0,a8->(1/2)-((((a4^2)+(a5^2)) (((a7 b4-a4 b7))^2))/(2 (((a4^2)+(a5^2)) b4 (a4+b4)+(a4^2) (a7+b7) b7))),b8->(1/2)+((3 (((a4^2)+(a5^2)) b4+a4 a7 b7) ((a4+b4) ((a4^2)+(a5^2)) b4+(a4^2) b7 (a7+b7)))/(2 a4 ((a4^2)+(a5^2)) (((a7 b4-a4 b7))^2)))}//Factor)=={(a5 (3 (a5^6) a7 (b4^6)+(a4^9) b4 (b7^5)+a4 (a5^4) a7 (b4^5) (6 (a5^2)+(a7^3) b7)+(a4^8) a7 (b4^2) (3 (b4^2)+3 (b7^2)-4 (b7^4))+(a4^2) (a5^4) a7 (b4^4) (3 (a5^2)+9 (b4^2)+b7 (6 a7+9 b7-4 (a7^2) b7))+(a4^6) a7 (3 (b4^6)+3 (b7^4) (((a7+b7))^2)+(b4^4) b7 (6 a7+9 b7-4 (a7^2) b7)+3 (b4^2) (b7^2) ((a7^2)+4 a7 b7+3 (b7^2))+(a5^2) (b4^2) (9 (b4^2)+6 (b7^2)-8 (b7^4)))+(a4^4) (a5^2) a7 (b4^2) (9 (b4^4)+2 (b4^2) b7 (6 a7+9 b7-4 (a7^2) b7)+3 (b7^2) ((a7^2)+4 a7 b7+3 (b7^2))+(a5^2) (9 (b4^2)+3 (b7^2)-4 (b7^4)))+2 (a4^3) ((a5^2) (a7^4) (b4^5) b7+3 (a5^4) a7 (b4^3) (3 (b4^2)+b7 (a7+2 b7+a7 (b7^2))))+(a4^5) b4 ((a7^4) (b4^4) b7+(a5^4) (b7^5)+6 (a5^2) a7 (3 (b4^4)+(b7^3) (a7+b7)+2 (b4^2) b7 (a7+2 b7+a7 (b7^2))))+2 (a4^7) b4 ((a5^2) (b7^5)+3 a7 ((((b4^2)+(b7^2)))^2)+3 (a7^2) ((b7^3)+(b4^2) (b7+(b7^3))))) sol141nsp1b6)/(2 (Sqrt[3]) a4 ((((a4^2)+(a5^2)))^2) (((a7 b4-a4 b7))^4) ((a4^2) b4+(a5^2) b4+a4 a7 b7) ((((a4^3) b4+a4 (a5^2) b4+(a5^2) (b4^2)+(a4^2) ((b4^2)+b7 (a7+b7))))^2)),-(((3 (a5^6) a7 (b4^6)+(a4^9) b4 (b7^5)+a4 (a5^4) a7 (b4^5) (6 (a5^2)+(a7^3) b7)+(a4^8) a7 (b4^2) (3 (b4^2)+3 (b7^2)-4 (b7^4))+(a4^2) (a5^4) a7 (b4^4) (3 (a5^2)+9 (b4^2)+b7 (6 a7+9 b7-4 (a7^2) b7))+(a4^6) a7 (3 (b4^6)+3 (b7^4) (((a7+b7))^2)+(b4^4) b7 (6 a7+9 b7-4 (a7^2) b7)+3 (b4^2) (b7^2) ((a7^2)+4 a7 b7+3 (b7^2))+(a5^2) (b4^2) (9 (b4^2)+6 (b7^2)-8 (b7^4)))+(a4^4) (a5^2) a7 (b4^2) (9 (b4^4)+2 (b4^2) b7 (6 a7+9 b7-4 (a7^2) b7)+3 (b7^2) ((a7^2)+4 a7 b7+3 (b7^2))+(a5^2) (9 (b4^2)+3 (b7^2)-4 (b7^4)))+2 (a4^3) ((a5^2) (a7^4) (b4^5) b7+3 (a5^4) a7 (b4^3) (3 (b4^2)+b7 (a7+2 b7+a7 (b7^2))))+(a4^5) b4 ((a7^4) (b4^4) b7+(a5^4) (b7^5)+6 (a5^2) a7 (3 (b4^4)+(b7^3) (a7+b7)+2 (b4^2) b7 (a7+2 b7+a7 (b7^2))))+2 (a4^7) b4 ((a5^2) (b7^5)+3 a7 ((((b4^2)+(b7^2)))^2)+3 (a7^2) ((b7^3)+(b4^2) (b7+(b7^3))))) sol141nsp1b6)/(2 (Sqrt[3]) ((((a4^2)+(a5^2)))^2) (((a7 b4-a4 b7))^4) ((a4^2) b4+(a5^2) b4+a4 a7 b7) ((((a4^3) b4+a4 (a5^2) b4+(a5^2) (b4^2)+(a4^2) ((b4^2)+b7 (a7+b7))))^2))),(sol141nsp1b6 ((a5^6) (3 (a5^2)-3 (a7^2)+(a7^4)) (b4^6)+2 a4 (a5^6) (b4^5) (3 (a5^2)-(a7^2) (3+2 a7 b7))+(a4^10) (3 (b4^4)-(b7^6)+(b4^2) (b7^2) (3+(b7^2)))+2 (a4^9) (3 (b4^5)+(b4^3) b7 (3 a7+6 b7-2 a7 (b7^2))+b4 (b7^3) (3 a7+3 b7+2 a7 (b7^2)))+(a4^2) (a5^4) (b4^4) (3 (a5^4)+(a7^2) (3 (-3+(a7^2)) (b4^2)-b7 (6 a7+9 b7+(a7^2) b7))+3 (a5^2) (4 (b4^2)+2 a7 b7+3 (b7^2)+(a7^2) (-1+2 (b7^2))))+2 (a4^3) (a5^4) (b4^3) ((a5^2) (12 (b4^2)+b7 (3 a7+6 b7-2 a7 (b7^2)))-(a7^2) ((b4^2) (9+6 a7 b7)+b7 (3 a7+6 b7-2 a7 (b7^2))))+2 (a4^5) (a5^2) b4 (-(a7^2) (3 (b7^3) (a7+b7)+(b4^4) (9+6 a7 b7)+2 (b4^2) b7 (3 a7+6 b7-2 a7 (b7^2)))+(a5^2) (18 (b4^4)+3 (b4^2) b7 (3 a7+6 b7-2 a7 (b7^2))+(b7^3) (3 a7+3 b7+2 a7 (b7^2))))+2 (a4^7) b4 (-(a7^2) (3 (b7^3) (a7+b7)+(b4^4) (3+2 a7 b7)+(b4^2) b7 (3 a7+6 b7-2 a7 (b7^2)))+(a5^2) (12 (b4^4)+3 (b4^2) b7 (3 a7+6 b7-2 a7 (b7^2))+2 (b7^3) (3 a7+3 b7+2 a7 (b7^2))))+(a4^8) ((a5^2) (12 (b4^4)-2 (b7^6)+3 (b4^2) (b7^2) (3+(b7^2)))+3 (2 a7 b7 ((((b4^2)+(b7^2)))^2)+((((b4^2)+(b7^2)))^3)+(a7^2) ((b7^4)-2 (b4^2) (b7^4)+(b4^4) (-1+2 (b7^2)))))+(a4^4) (a5^2) (b4^2) ((a5^4) (12 (b4^2)+3 (b7^2)+(b7^4))+(a7^2) (3 (-3+(a7^2)) (b4^4)-2 (b4^2) b7 (6 a7+9 b7+(a7^2) b7)-3 (b7^2) ((a7^2)+4 a7 b7+3 (b7^2)))+3 (a5^2) (a7 (6 (b4^2) b7+4 (b7^3))+3 (2 (b4^4)+3 (b4^2) (b7^2)+(b7^4))+(a7^2) (-2 (b7^4)+(b4^2) (-3+6 (b7^2)))))+(a4^6) ((a5^4) (18 (b4^4)-(b7^6)+3 (b4^2) (b7^2) (3+(b7^2)))+(a7^2) ((-3+(a7^2)) (b4^6)-3 (b7^4) (((a7+b7))^2)-(b4^4) b7 (6 a7+9 b7+(a7^2) b7)-3 (b4^2) (b7^2) ((a7^2)+4 a7 b7+3 (b7^2)))+3 (a5^2) (((((b4^2)+(b7^2)))^2) (4 (b4^2)+(b7^2))+2 a7 (3 (b4^4) b7+4 (b4^2) (b7^3)+(b7^5))+(a7^2) ((b7^4)-4 (b4^2) (b7^4)+(b4^4) (-3+6 (b7^2)))))))/(4 (Sqrt[3]) a4 ((((a4^2)+(a5^2)))^2) (((a7 b4-a4 b7))^4) ((a4^2) b4+(a5^2) b4+a4 a7 b7) ((((a4^3) b4+a4 (a5^2) b4+(a5^2) (b4^2)+(a4^2) ((b4^2)+b7 (a7+b7))))^2)),(a4 b7 sol141nsp1b6)/(4 ((a4^2)+(a5^2)) (-a7 b4+a4 b7) ((a4^2) b4+(a5^2) b4+a4 a7 b7) ((a4^3) b4+a4 (a5^2) b4+(a5^2) (b4^2)+(a4^2) ((b4^2)+b7 (a7+b7)))),(a5 b7 sol141nsp1b6)/(4 ((a4^2)+(a5^2)) (-a7 b4+a4 b7) ((a4^2) b4+(a5^2) b4+a4 a7 b7) ((a4^3) b4+a4 (a5^2) b4+(a5^2) (b4^2)+(a4^2) ((b4^2)+b7 (a7+b7)))),0,-((b4 sol141nsp1b6)/(4 a4 (-a7 b4+a4 b7) ((a4^2) b4+(a5^2) b4+a4 a7 b7) ((a4^3) b4+a4 (a5^2) b4+(a5^2) (b4^2)+(a4^2) ((b4^2)+b7 (a7+b7))))),(((a5^2) (b4^2)+(a4^2) ((b4^2)+(b7^2))) sol141nsp1b6)/(4 a4 ((a4^2)+(a5^2)) (((a7 b4-a4 b7))^2) ((a4^2) b4+(a5^2) b4+a4 a7 b7) ((a4^3) b4+a4 (a5^2) b4+(a5^2) (b4^2)+(a4^2) ((b4^2)+b7 (a7+b7))))}//Simplify

eqd7a/.{b5->a5 b4/a4,a6->0,b6->0}//Simplify

eqd7a6/.{b5->a5 b4/a4,a6->0,b6->0}//Simplify

Ce cas n’émarge pas de eqd7a pi eqd7a6 mais du plus général eq171na

(eq171na/.{b5->a5 b4/a4,a6->0,b6->0})==(1/(a4^6))(a5^2) a7 (((a7 b4+a4 b7))^2) sol141nsp1b6//Simplify

(sol478nou/.{b5->a5 b4/a4,a6->0,b6->0})==(1/(a4^5))((a4^2) b4+(a5^2) b4+a4 a7 b7)sol141nsp1b6//Simplify

solution a4=b4=a6=b6=0.

Restent quatre variables (a5,b5,a7,b7) avec la règle

Length[{a1->(((Sqrt[3]) a5 a7 (a5 b5+(b5^2)+b7 (a7+b7)))/(((a7 b5-a5 b7))^2)),b1->-((b5 b7 (((a7 b5-a5 b7))^2))/((Sqrt[3]) ((a5^2) (b5^2)+a7 b7 ((b5^2)+b7 (a7+b7))+a5 ((b5^3)+b5 b7 (2 a7+b7))))),a2->0,b2->0,a3->(((Sqrt[3]) ((a5^2)-(a7^2)) (a5 b5+(b5^2)+b7 (a7+b7)))/(2 (((a7 b5-a5 b7))^2))),b3->-(((((a7 b5-a5 b7))^2) ((b5^2)-(b7^2)))/(2 (Sqrt[3]) ((a5^2) (b5^2)+a7 b7 ((b5^2)+b7 (a7+b7))+a5 ((b5^3)+b5 b7 (2 a7+b7))))),a4->0,b4->0,a6->0,b6->0,a8->(1/2)-((((a7 b5-a5 b7))^2)/(2 (b5 (a5+b5)+b7 (a7+b7)))),b8->(1/2)+((3 (a5 b5+a7 b7) (b5 (a5+b5)+b7 (a7+b7)))/(2 (((a7 b5-a5 b7))^2)))}]

L’équation est sol4868sp1a4

sol4868sp1a4

(cc[{a1,a2,a3,a4,a5,a6,a7,a8},{b1,b2,b3,b4,b5,b6,b7,b8}]+{a1,a2,a3,a4,a5,a6,a7,a8}+{b1,b2,b3,b4,b5,b6,b7,b8}/.{a1->(((Sqrt[3]) a5 a7 (a5 b5+(b5^2)+b7 (a7+b7)))/(((a7 b5-a5 b7))^2)),b1->-((b5 b7 (((a7 b5-a5 b7))^2))/((Sqrt[3]) ((a5^2) (b5^2)+a7 b7 ((b5^2)+b7 (a7+b7))+a5 ((b5^3)+b5 b7 (2 a7+b7))))),a2->0,b2->0,a3->(((Sqrt[3]) ((a5^2)-(a7^2)) (a5 b5+(b5^2)+b7 (a7+b7)))/(2 (((a7 b5-a5 b7))^2))),b3->-(((((a7 b5-a5 b7))^2) ((b5^2)-(b7^2)))/(2 (Sqrt[3]) ((a5^2) (b5^2)+a7 b7 ((b5^2)+b7 (a7+b7))+a5 ((b5^3)+b5 b7 (2 a7+b7))))),a4->0,b4->0,a6->0,b6->0,a8->(1/2)-((((a7 b5-a5 b7))^2)/(2 (b5 (a5+b5)+b7 (a7+b7)))),b8->(1/2)+((3 (a5 b5+a7 b7) (b5 (a5+b5)+b7 (a7+b7)))/(2 (((a7 b5-a5 b7))^2)))}//Factor)=={((3 a5 a7 ((b5^2)+(b7^2)) (((b5 (a5+b5)+b7 (a7+b7)))^2)+b5 b7 (((a7 b5-a5 b7))^4)) sol4868sp1a4)/(2 (Sqrt[3]) (((a7 b5-a5 b7))^4) (a5 b5+a7 b7) (((a5 b5+(b5^2)+b7 (a7+b7)))^2)),0,(sol4868sp1a4 ((((a7 b5-a5 b7))^4) ((b5^2)-(b7^2))+6 ((a5^2)-(a7^2)) (a5 b5+a7 b7) ((((b5^2)+(b7^2)))^2)+3 ((a5^2)-(a7^2)) (((a5 b5+a7 b7))^2) ((b5^2)+(b7^2))+3 ((a5^2)-(a7^2)) ((((b5^2)+(b7^2)))^3)))/(4 (Sqrt[3]) (((a7 b5-a5 b7))^4) (a5 b5+a7 b7) (((a5 b5+(b5^2)+b7 (a7+b7)))^2)),0,-((b7 sol4868sp1a4)/(4 (a7 b5-a5 b7) (a5 b5+a7 b7) (a5 b5+(b5^2)+b7 (a7+b7)))),0,(b5 sol4868sp1a4)/(4 (a7 b5-a5 b7) (a5 b5+a7 b7) (a5 b5+(b5^2)+b7 (a7+b7))),(((b5^2)+(b7^2)) sol4868sp1a4)/(4 (((a7 b5-a5 b7))^2) (a5 b5+a7 b7) (a5 b5+(b5^2)+b7 (a7+b7)))}//Simplify

({a1,a2,a3,a4,a5,a6,a7,a8}.{b1,b2,b3,b4,b5,b6,b7,b8}+1/2/.{a1->(((Sqrt[3]) a5 a7 (a5 b5+(b5^2)+b7 (a7+b7)))/(((a7 b5-a5 b7))^2)),b1->-((b5 b7 (((a7 b5-a5 b7))^2))/((Sqrt[3]) ((a5^2) (b5^2)+a7 b7 ((b5^2)+b7 (a7+b7))+a5 ((b5^3)+b5 b7 (2 a7+b7))))),a2->0,b2->0,a3->(((Sqrt[3]) ((a5^2)-(a7^2)) (a5 b5+(b5^2)+b7 (a7+b7)))/(2 (((a7 b5-a5 b7))^2))),b3->-(((((a7 b5-a5 b7))^2) ((b5^2)-(b7^2)))/(2 (Sqrt[3]) ((a5^2) (b5^2)+a7 b7 ((b5^2)+b7 (a7+b7))+a5 ((b5^3)+b5 b7 (2 a7+b7))))),a4->0,b4->0,a6->0,b6->0,a8->(1/2)-((((a7 b5-a5 b7))^2)/(2 (b5 (a5+b5)+b7 (a7+b7)))),b8->(1/2)+((3 (a5 b5+a7 b7) (b5 (a5+b5)+b7 (a7+b7)))/(2 (((a7 b5-a5 b7))^2)))}//Factor)==((b5^2)+(b7^2)) sol4868sp1a4/(4 (((a7 b5-a5 b7))^2) (a5 b5+a7 b7) (a5 b5+(b5^2)+b7 (a7+b7)))//Simplify

(eq1a/.{a1->(((Sqrt[3]) a5 a7 (a5 b5+(b5^2)+b7 (a7+b7)))/(((a7 b5-a5 b7))^2)),b1->-((b5 b7 (((a7 b5-a5 b7))^2))/((Sqrt[3]) ((a5^2) (b5^2)+a7 b7 ((b5^2)+b7 (a7+b7))+a5 ((b5^3)+b5 b7 (2 a7+b7))))),a2->0,b2->0,a3->(((Sqrt[3]) ((a5^2)-(a7^2)) (a5 b5+(b5^2)+b7 (a7+b7)))/(2 (((a7 b5-a5 b7))^2))),b3->-(((((a7 b5-a5 b7))^2) ((b5^2)-(b7^2)))/(2 (Sqrt[3]) ((a5^2) (b5^2)+a7 b7 ((b5^2)+b7 (a7+b7))+a5 ((b5^3)+b5 b7 (2 a7+b7))))),a4->0,b4->0,a6->0,b6->0,a8->(1/2)-((((a7 b5-a5 b7))^2)/(2 (b5 (a5+b5)+b7 (a7+b7)))),b8->(1/2)+((3 (a5 b5+a7 b7) (b5 (a5+b5)+b7 (a7+b7)))/(2 (((a7 b5-a5 b7))^2)))}//Factor)==(sol4868sp1a4/(3 (((a5 b5+(b5^2)+b7 (a7+b7)))^2)))//Simplify

(eq1b/.{a1->(((Sqrt[3]) a5 a7 (a5 b5+(b5^2)+b7 (a7+b7)))/(((a7 b5-a5 b7))^2)),b1->-((b5 b7 (((a7 b5-a5 b7))^2))/((Sqrt[3]) ((a5^2) (b5^2)+a7 b7 ((b5^2)+b7 (a7+b7))+a5 ((b5^3)+b5 b7 (2 a7+b7))))),a2->0,b2->0,a3->(((Sqrt[3]) ((a5^2)-(a7^2)) (a5 b5+(b5^2)+b7 (a7+b7)))/(2 (((a7 b5-a5 b7))^2))),b3->-(((((a7 b5-a5 b7))^2) ((b5^2)-(b7^2)))/(2 (Sqrt[3]) ((a5^2) (b5^2)+a7 b7 ((b5^2)+b7 (a7+b7))+a5 ((b5^3)+b5 b7 (2 a7+b7))))),a4->0,b4->0,a6->0,b6->0,a8->(1/2)-((((a7 b5-a5 b7))^2)/(2 (b5 (a5+b5)+b7 (a7+b7)))),b8->(1/2)+((3 (a5 b5+a7 b7) (b5 (a5+b5)+b7 (a7+b7)))/(2 (((a7 b5-a5 b7))^2)))}//Factor)==(((b5^2)+(b7^2))/(((a7 b5-a5 b7))^4))sol4868sp1a4//Simplify

eqd7a/.{a4->0,b4->0,a6->0,b6->0}

eqd7a6/.{a4->0,b4->0,a6->0,b6->0}

Ce cas n’émarge pas de eqd7a pi eqd7a6 mais du plus général eq171na ou de sol478nou (mais pas de sol478na, ce qui m’a trompé).

eq171na==a7 (((a7 b5+a5 b7))^2)sol4868sp1a4/.{a4->0,b4->0,a6->0,b6->0}//Simplify

(sol478nou/.{a4->0,b4->0,a6->0,b6->0})==(a5 b5+a7 b7)sol4868sp1a4//Simplify

solution a6=b6=0 avec a7=b7

Restent  variables (a4,a5,a7) avec la règle

Length[{a1->((2 a5 ((a4^2)+(a5^2)) a7)/((Sqrt[3]) ((a4^2)+(a5^2)-(a7^2)))),b1->-((2 a5 ((a4^2)+(a5^2)) a7)/((Sqrt[3]) ((a4^2)+(a5^2)-(a7^2)))),a2->-((2 a4 ((a4^2)+(a5^2)) a7)/((Sqrt[3]) ((a4^2)+(a5^2)-(a7^2)))),b2->((2 a4 ((a4^2)+(a5^2)) a7)/((Sqrt[3]) ((a4^2)+(a5^2)-(a7^2)))),a3->(((a4^2)+(a5^2))/(Sqrt[3])),b3->(((a4^2)+(a5^2))/(Sqrt[3])),b4->-a4,b5->-a5,a6->0,b6->0,b7->a7,a8->-(1/4)+(3/4)((a7^2)/((a4^2)+(a5^2))),b8->-(1/4)+(3/4)((a7^2)/((a4^2)+(a5^2)))}]

L’équation est  (a7^2)=((a4^2)+(a5^2))(1-(4/3)((a4^2)+(a5^2)))

(cc[{a1,a2,a3,a4,a5,a6,a7,a8},{b1,b2,b3,b4,b5,b6,b7,b8}]+{a1,a2,a3,a4,a5,a6,a7,a8}+{b1,b2,b3,b4,b5,b6,b7,b8}/.{a1->((2 a5 ((a4^2)+(a5^2)) a7)/((Sqrt[3]) ((a4^2)+(a5^2)-(a7^2)))),b1->-((2 a5 ((a4^2)+(a5^2)) a7)/((Sqrt[3]) ((a4^2)+(a5^2)-(a7^2)))),a2->-((2 a4 ((a4^2)+(a5^2)) a7)/((Sqrt[3]) ((a4^2)+(a5^2)-(a7^2)))),b2->((2 a4 ((a4^2)+(a5^2)) a7)/((Sqrt[3]) ((a4^2)+(a5^2)-(a7^2)))),a3->(((a4^2)+(a5^2))/(Sqrt[3])),b3->(((a4^2)+(a5^2))/(Sqrt[3])),b4->-a4,b5->-a5,a6->0,b6->0,b7->a7,a8->-(1/4)+(3/4)((a7^2)/((a4^2)+(a5^2))),b8->-(1/4)+(3/4)((a7^2)/((a4^2)+(a5^2)))}//Factor)=={0,0,0,0,0,0,((a7 (-3 (a4^2)-3 (a5^2)+(a7^2)) sol7a5a6)/(4 ((a4^2)+(a5^2)) ((a4^2)+(a5^2)-(a7^2)))),(sol7a5a6/(48 ((((a4^2)+(a5^2)))^2) ((((a4^2)+(a5^2)-(a7^2)))^2))) (((((a4^2)+(a5^2)))^3) (9+4 (a4^2)+4 (a5^2))-3 ((((a4^2)+(a5^2)))^2) (13+8 (a4^2)+8 (a5^2)) (a7^2)+((a4^2)+(a5^2)) (39+4 (a4^2)+4 (a5^2)) (a7^4)-9 (a7^6))}//Simplify

({a1,a2,a3,a4,a5,a6,a7,a8}.{b1,b2,b3,b4,b5,b6,b7,b8}+1/2/.{a1->((2 a5 ((a4^2)+(a5^2)) a7)/((Sqrt[3]) ((a4^2)+(a5^2)-(a7^2)))),b1->-((2 a5 ((a4^2)+(a5^2)) a7)/((Sqrt[3]) ((a4^2)+(a5^2)-(a7^2)))),a2->-((2 a4 ((a4^2)+(a5^2)) a7)/((Sqrt[3]) ((a4^2)+(a5^2)-(a7^2)))),b2->((2 a4 ((a4^2)+(a5^2)) a7)/((Sqrt[3]) ((a4^2)+(a5^2)-(a7^2)))),a3->(((a4^2)+(a5^2))/(Sqrt[3])),b3->(((a4^2)+(a5^2))/(Sqrt[3])),b4->-a4,b5->-a5,a6->0,b6->0,b7->a7,a8->-(1/4)+(3/4)((a7^2)/((a4^2)+(a5^2))),b8->-(1/4)+(3/4)((a7^2)/((a4^2)+(a5^2)))}//Factor)==(sol7a5a6/(48 ((((a4^2)+(a5^2)))^2) ((((a4^2)+(a5^2)-(a7^2)))^2))) (((((a4^2)+(a5^2)))^3) (-9+4 (a4^2)+4 (a5^2))-3 ((((a4^2)+(a5^2)))^2) (-5+8 (a4^2)+8 (a5^2)) (a7^2)+((a4^2)+(a5^2)) (-15+4 (a4^2)+4 (a5^2)) (a7^4)+9 (a7^6))//Simplify

(eq1a/.{a1->((2 a5 ((a4^2)+(a5^2)) a7)/((Sqrt[3]) ((a4^2)+(a5^2)-(a7^2)))),b1->-((2 a5 ((a4^2)+(a5^2)) a7)/((Sqrt[3]) ((a4^2)+(a5^2)-(a7^2)))),a2->-((2 a4 ((a4^2)+(a5^2)) a7)/((Sqrt[3]) ((a4^2)+(a5^2)-(a7^2)))),b2->((2 a4 ((a4^2)+(a5^2)) a7)/((Sqrt[3]) ((a4^2)+(a5^2)-(a7^2)))),a3->(((a4^2)+(a5^2))/(Sqrt[3])),b3->(((a4^2)+(a5^2))/(Sqrt[3])),b4->-a4,b5->-a5,a6->0,b6->0,b7->a7,a8->-(1/4)+(3/4)((a7^2)/((a4^2)+(a5^2))),b8->-(1/4)+(3/4)((a7^2)/((a4^2)+(a5^2)))}//Factor)==((((a4^2)+(a5^2)+(a7^2)) sol7a5a6)/(4 ((((a4^2)+(a5^2)))^2)))//Simplify

(eq1b/.{a1->((2 a5 ((a4^2)+(a5^2)) a7)/((Sqrt[3]) ((a4^2)+(a5^2)-(a7^2)))),b1->-((2 a5 ((a4^2)+(a5^2)) a7)/((Sqrt[3]) ((a4^2)+(a5^2)-(a7^2)))),a2->-((2 a4 ((a4^2)+(a5^2)) a7)/((Sqrt[3]) ((a4^2)+(a5^2)-(a7^2)))),b2->((2 a4 ((a4^2)+(a5^2)) a7)/((Sqrt[3]) ((a4^2)+(a5^2)-(a7^2)))),a3->(((a4^2)+(a5^2))/(Sqrt[3])),b3->(((a4^2)+(a5^2))/(Sqrt[3])),b4->-a4,b5->-a5,a6->0,b6->0,b7->a7,a8->-(1/4)+(3/4)((a7^2)/((a4^2)+(a5^2))),b8->-(1/4)+(3/4)((a7^2)/((a4^2)+(a5^2)))}//Factor)==((((a4^2)+(a5^2)+(a7^2)) sol7a5a6)/(4 ((((a4^2)+(a5^2)))^2)))//Simplify

Cette solution est originale et dérive de sol478na ou mieux, de sol478nou

(sol478na/.{b4->-a4,b5->-a5,a6->0,b6->0,b7->a7}//Factor)==-8 a4 (a7^5) ((a4^2)+(a5^2)-(a7^2))sol7a5a6//Simplify

(sol478nou/.{b4->-a4,b5->-a5,a6->0,b6->0,b7->a7})==4 (a7^4) (-(a4^2)-(a5^2)+(a7^2))sol7a5a6//Simplify

solution a5=b5=a6=b6=0 (premier cas, b7+a7=0)

Restent trois  variables (a4,a7,a8) avec la règle

Length[{a1->0,b1->0,a2->(((Sqrt[3]) a4 a7)/(-1+2 a8)),b2->-(((Sqrt[3]) a4 a7)/(2+2 a8)),a3->(((Sqrt[3]) (-(a4^2)+(a7^2)))/(-2+4 a8)),b3->(((Sqrt[3]) ((a4^2)-(a7^2)))/(4 (1+a8))),b4->-a4, a5->0,b5->0, a6->0, b6->0,b7->-a7,b8->-(1/2)-a8}]

L’équation est eq1a56, c’est eq1a (avec a5=a6=0)

eq1a56==3(eq1a/.{a5->0,a6->0})//Simplify

(eq1a/.{a5->0,a6->0})

(cc[{a1,a2,a3,a4,a5,a6,a7,a8},{b1,b2,b3,b4,b5,b6,b7,b8}]+{a1,a2,a3,a4,a5,a6,a7,a8}+{b1,b2,b3,b4,b5,b6,b7,b8}/.{a1->0,b1->0,a2->(((Sqrt[3]) a4 a7)/(-1+2 a8)),b2->-(((Sqrt[3]) a4 a7)/(2+2 a8)),a3->(((Sqrt[3]) (-(a4^2)+(a7^2)))/(-2+4 a8)),b3->(((Sqrt[3]) ((a4^2)-(a7^2)))/(4 (1+a8))),b4->-a4, a5->0,b5->0, a6->0, b6->0,b7->-a7,b8->-(1/2)-a8}//Factor)=={0,0,0,((a4 (1+4 a8) eq1a56)/(8 (1+a8) (-1+2 a8))),0,0,((a7 (1+4 a8) eq1a56)/(8 (1+a8) (-1+2 a8))),-(((2+(a4^2)+(a7^2)-2 a8-4 (a8^2)) eq1a56)/(8 (1+a8) (-1+2 a8)))}//Simplify

({a1,a2,a3,a4,a5,a6,a7,a8}.{b1,b2,b3,b4,b5,b6,b7,b8}+1/2/.{a1->0,b1->0,a2->(((Sqrt[3]) a4 a7)/(-1+2 a8)),b2->-(((Sqrt[3]) a4 a7)/(2+2 a8)),a3->(((Sqrt[3]) (-(a4^2)+(a7^2)))/(-2+4 a8)),b3->(((Sqrt[3]) ((a4^2)-(a7^2)))/(4 (1+a8))),b4->-a4, a5->0,b5->0, a6->0, b6->0,b7->-a7,b8->-(1/2)-a8}//Factor)==-(((-2+(a4^2)+(a7^2)+2 a8+4 (a8^2))  eq1a56)/(8 (-1+a8+2 (a8^2))))//Simplify

Je ne fais pas eq1a ici qui est triviale (et eq1b presque)

(eq1b/.{a1->0,b1->0,a2->(((Sqrt[3]) a4 a7)/(-1+2 a8)),b2->-(((Sqrt[3]) a4 a7)/(2+2 a8)),a3->(((Sqrt[3]) (-(a4^2)+(a7^2)))/(-2+4 a8)),b3->(((Sqrt[3]) ((a4^2)-(a7^2)))/(4 (1+a8))),b4->-a4, a5->0,b5->0, a6->0, b6->0,b7->-a7,b8->-(1/2)-a8}//Factor)==(eq1a56/3)//Simplify

N' appartient à aucune des équations courantes. Normalement, reg8na8 est toujours valable mais ici con8nr=0

con8nr/.{b4->-a4, a5->0,b5->0, a6->0, b6->0,b7->-a7}//Simplify

On sait que lorsque con8nr=0 et con88b≠0, l’équation se reporte immédiatement sur cnu8n mais ça ne donne rien ici

con88b/.{b4->-a4, a5->0,b5->0, a6->0, b6->0,b7->-a7}//Simplify

cnu8n/.{b4->-a4, a5->0,b5->0, a6->0, b6->0,b7->-a7}//Simplify

On peut essayer avec con1nr puis cnu1n (ou con2nr etc.) mais ça ne donne rien finalement

con1nr/.{b4->-a4, a5->0,b5->0, a6->0, b6->0,b7->-a7}//Simplify

cnu1n/.{b4->-a4, a5->0,b5->0, a6->0, b6->0,b7->-a7}//Simplify

con2nr/.{b4->-a4, a5->0,b5->0, a6->0, b6->0,b7->-a7}//Simplify

cnu2n/.{b4->-a4, a5->0,b5->0, a6->0, b6->0,b7->-a7}//Simplify

con3nr/.{b4->-a4, a5->0,b5->0, a6->0, b6->0,b7->-a7}//Simplify

cnu3n/.{b4->-a4, a5->0,b5->0, a6->0, b6->0,b7->-a7}//Simplify

pourtant, on peut recalculer reg8na8 pour ce cas particulier et ça donne le bon résultat ; on restreint reg88 et regn8 sur ce cas :

(reg88/.{b4->-a4, a5->0,b5->0, a6->0, b6->0,b7->-a7})==reg8856//Simplify

(regn8/.{b4->-a4, a5->0,b5->0, a6->0, b6->0,b7->-a7})==regn856//Simplify

reg8856[[1,2]]-regn856[[1,2]]==((9((((a4^2)+(a7^2)))^2) eq1a56)/(2(3-6 a8+((a4^2)+(a7^2)) (-5+4 a8))(6-12 a8+((a4^2)+(a7^2)) (-7+2 a8))))//Simplify

mais ce cas reste séparé car inatteignable par les équations générales. J’ai supprimé d’autres cas donnant la même eq1a56 mais dérivée des équations générales. Ce ne sont pas les équations qui sont fragiles mais les paramétrages, celui-ci est très particulier. Toutefois, c’est un sous-cas de la solution 7

solution a5=b5=a7=b7=0

Restent deux  variables (a4,a6) avec la règle

Length[{a1->((2 a4 a6)/(Sqrt[3])),b1->((2 a4 a6)/(Sqrt[3])),a2->0,b2->0,a3->(((a4^2)-(a6^2))/(Sqrt[3])),b3->(((a4^2)-(a6^2))/(Sqrt[3])),b4->-a4, a5->0,b5->0,b6->-a6,a7->0,b7->0,a8->-(1/4),b8->-(1/4)}]

L’équation est S2 (a4^2)+(a6^2)=(3/4)

(cc[{a1,a2,a3,a4,a5,a6,a7,a8},{b1,b2,b3,b4,b5,b6,b7,b8}]+{a1,a2,a3,a4,a5,a6,a7,a8}+{b1,b2,b3,b4,b5,b6,b7,b8}/.{a1->((2 a4 a6)/(Sqrt[3])),b1->((2 a4 a6)/(Sqrt[3])),a2->0,b2->0,a3->(((a4^2)-(a6^2))/(Sqrt[3])),b3->(((a4^2)-(a6^2))/(Sqrt[3])),b4->-a4, a5->0,b5->0,b6->-a6,a7->0,b7->0,a8->-(1/4),b8->-(1/4)}//Factor)

({a1,a2,a3,a4,a5,a6,a7,a8}.{b1,b2,b3,b4,b5,b6,b7,b8}+1/2/.{a1->((2 a4 a6)/(Sqrt[3])),b1->((2 a4 a6)/(Sqrt[3])),a2->0,b2->0,a3->(((a4^2)-(a6^2))/(Sqrt[3])),b3->(((a4^2)-(a6^2))/(Sqrt[3])),b4->-a4, a5->0,b5->0,b6->-a6,a7->0,b7->0,a8->-(1/4),b8->-(1/4)}//Factor)

(eq1a/.{a1->((2 a4 a6)/(Sqrt[3])),b1->((2 a4 a6)/(Sqrt[3])),a2->0,b2->0,a3->(((a4^2)-(a6^2))/(Sqrt[3])),b3->(((a4^2)-(a6^2))/(Sqrt[3])),b4->-a4, a5->0,b5->0,b6->-a6,a7->0,b7->0,a8->-(1/4),b8->-(1/4)}//Factor)

(eq1b/.{a1->((2 a4 a6)/(Sqrt[3])),b1->((2 a4 a6)/(Sqrt[3])),a2->0,b2->0,a3->(((a4^2)-(a6^2))/(Sqrt[3])),b3->(((a4^2)-(a6^2))/(Sqrt[3])),b4->-a4, a5->0,b5->0,b6->-a6,a7->0,b7->0,a8->-(1/4),b8->-(1/4)}//Factor)

Ce cas émarge du cas plus général dont l’équation est simplement eq1a avec a5=a6=0

((cc[{a1,a2,a3,a4,a5,a6,a7,a8},{b1,b2,b3,b4,b5,b6,b7,b8}]+{a1,a2,a3,a4,a5,a6,a7,a8}+{b1,b2,b3,b4,b5,b6,b7,b8}/.{a1->-(((Sqrt[3]) a4 a6)/(-1+2 a8)),b1->-((a4 a6 (-1+2 a8))/((Sqrt[3]) ((a4^2)+(a6^2)))),a2->0,b2->0,a3->-(((Sqrt[3]) (a4-a6) (a4+a6))/(2 (-1+2 a8))),b3->-(((a4-a6) (a4+a6) (-1+2 a8))/(2 (Sqrt[3]) ((a4^2)+(a6^2)))),b4->-a4, a5->0,b5->0,b6->-a6,a7->0,b7->0,b8->(1/2)+((3 ((a4^2)+(a6^2)))/(-2+4 a8))}//Factor)/.{(a8^2)->(1/4) (2-3 (a4^2)-3 (a6^2)-2 a8)}//Factor)//Factor

(({a1,a2,a3,a4,a5,a6,a7,a8}.{b1,b2,b3,b4,b5,b6,b7,b8}+1/2/.{a1->-(((Sqrt[3]) a4 a6)/(-1+2 a8)),b1->-((a4 a6 (-1+2 a8))/((Sqrt[3]) ((a4^2)+(a6^2)))),a2->0,b2->0,a3->-(((Sqrt[3]) (a4-a6) (a4+a6))/(2 (-1+2 a8))),b3->-(((a4-a6) (a4+a6) (-1+2 a8))/(2 (Sqrt[3]) ((a4^2)+(a6^2)))),b4->-a4, a5->0,b5->0,b6->-a6,a7->0,b7->0,b8->(1/2)+((3 ((a4^2)+(a6^2)))/(-2+4 a8))}//Factor)/.{(a8^2)->(1/4) (2-3 (a4^2)-3 (a6^2)-2 a8)}//Factor)//Factor

((eq1a/.{a1->-(((Sqrt[3]) a4 a6)/(-1+2 a8)),b1->-((a4 a6 (-1+2 a8))/((Sqrt[3]) ((a4^2)+(a6^2)))),a2->0,b2->0,a3->-(((Sqrt[3]) (a4-a6) (a4+a6))/(2 (-1+2 a8))),b3->-(((a4-a6) (a4+a6) (-1+2 a8))/(2 (Sqrt[3]) ((a4^2)+(a6^2)))),b4->-a4, a5->0,b5->0,b6->-a6,a7->0,b7->0,b8->(1/2)+((3 ((a4^2)+(a6^2)))/(-2+4 a8))}//Factor)/.{(a8^2)->(1/4) (2-3 (a4^2)-3 (a6^2)-2 a8)}//Factor)//Factor

((eq1b/.{a1->-(((Sqrt[3]) a4 a6)/(-1+2 a8)),b1->-((a4 a6 (-1+2 a8))/((Sqrt[3]) ((a4^2)+(a6^2)))),a2->0,b2->0,a3->-(((Sqrt[3]) (a4-a6) (a4+a6))/(2 (-1+2 a8))),b3->-(((a4-a6) (a4+a6) (-1+2 a8))/(2 (Sqrt[3]) ((a4^2)+(a6^2)))),b4->-a4, a5->0,b5->0,b6->-a6,a7->0,b7->0,b8->(1/2)+((3 ((a4^2)+(a6^2)))/(-2+4 a8))}//Factor)/.{(a8^2)->(1/4) (2-3 (a4^2)-3 (a6^2)-2 a8)}//Factor)//Factor

reg18[[1,2]]-regn8[[1,2]]/.{b4->-a4, a5->0,b5->0,b6->-a6,a7->0,b7->0}//Simplify

C’est un sous-cas de la solution 7 et la voie de con1nr=0 est également inutile (je note que le résultat obtenu par reg1na8 EST FAUX, ce qui est normal puisque con1nr=0).

con1nr/.{b4->-a4, a5->0,b5->0,b6->-a6,a7->0,b7->0}//Simplify

con88b/.{b4->-a4, a5->0,b5->0,b6->-a6,a7->0,b7->0}//Simplify

cnu1n/.{b4->-a4, a5->0,b5->0,b6->-a6,a7->0,b7->0}//Simplify

reg1na8/.{b4->-a4, a5->0,b5->0,b6->-a6,a7->0,b7->0}

solution a4=b4=a5=a6=b6=0

Restent trois  variables (b5,a7,b7) avec la règle

Length[{a1->0,b1->(((Sqrt[3]) b7 (a7+b7))/(a7 b5)),a2->0,b2->0,a3->-(((Sqrt[3]) ((b5^2)+b7 (a7+b7)))/(2 (b5^2))),b3->(((Sqrt[3]) (a7+b7) ((b5^2)-(b7^2)))/(2 a7 (b5^2))),b4->0,a4->0,a5->0,b6->0,a6->0,a8->(1/2)-(((a7^2) (b5^2))/(2 ((b5^2)+a7 b7+(b7^2)))),b8->(1/2)-((a7 (b5^2))/(2 (a7+b7)))}]

L’équation est  solpra1

solpra1

(cc[{a1,a2,a3,a4,a5,a6,a7,a8},{b1,b2,b3,b4,b5,b6,b7,b8}]+{a1,a2,a3,a4,a5,a6,a7,a8}+{b1,b2,b3,b4,b5,b6,b7,b8}/.{a1->0,b1->(((Sqrt[3]) b7 (a7+b7))/(a7 b5)),a2->0,b2->0,a3->-(((Sqrt[3]) ((b5^2)+b7 (a7+b7)))/(2 (b5^2))),b3->(((Sqrt[3]) (a7+b7) ((b5^2)-(b7^2)))/(2 a7 (b5^2))),b4->0,a4->0,a5->0,b6->0,a6->0,a8->(1/2)-(((a7^2) (b5^2))/(2 ((b5^2)+a7 b7+(b7^2)))),b8->(1/2)-((a7 (b5^2))/(2 (a7+b7)))}//Factor)=={(((Sqrt[3]) solpra1)/(2 a7 b5 ((b5^2)+b7 (a7+b7)))),0,-(((Sqrt[3]) (a7-b5+b7) (a7+b5+b7) solpra1)/(4 a7 (b5^2) (a7+b7) ((b5^2)+b7 (a7+b7)))),0,(solpra1/(4 ((b5^3)+b5 b7 (a7+b7)))),0,((((b5^2)+2 b7 (a7+b7)) solpra1)/(4 (b5^2) (a7+b7) ((b5^2)+b7 (a7+b7)))),-((((b5^4)-(b7^4)+a7 b7 ((b5^2)-2 (b7^2))+(a7^2) ((b5^2)+(b5^4)-(b7^2))) solpra1)/(4 a7 (b5^4) (a7+b7) ((b5^2)+b7 (a7+b7))))}//Simplify

({a1,a2,a3,a4,a5,a6,a7,a8}.{b1,b2,b3,b4,b5,b6,b7,b8}+1/2/.{a1->0,b1->(((Sqrt[3]) b7 (a7+b7))/(a7 b5)),a2->0,b2->0,a3->-(((Sqrt[3]) ((b5^2)+b7 (a7+b7)))/(2 (b5^2))),b3->(((Sqrt[3]) (a7+b7) ((b5^2)-(b7^2)))/(2 a7 (b5^2))),b4->0,a4->0,a5->0,b6->0,a6->0,a8->(1/2)-(((a7^2) (b5^2))/(2 ((b5^2)+a7 b7+(b7^2)))),b8->(1/2)-((a7 (b5^2))/(2 (a7+b7)))}//Factor)==((solpra1 (-(b5^4)+(b7^4)+(a7^2) (-(b5^2)+(b5^4)+(b7^2))+a7 (-(b5^2) b7+2 (b7^3))))/(4 a7 (b5^4) (a7+b7) ((b5^2)+b7 (a7+b7))))//Simplify

(eq1a/.{a1->0,b1->(((Sqrt[3]) b7 (a7+b7))/(a7 b5)),a2->0,b2->0,a3->-(((Sqrt[3]) ((b5^2)+b7 (a7+b7)))/(2 (b5^2))),b3->(((Sqrt[3]) (a7+b7) ((b5^2)-(b7^2)))/(2 a7 (b5^2))),b4->0,a4->0,a5->0,b6->0,a6->0,a8->(1/2)-(((a7^2) (b5^2))/(2 ((b5^2)+a7 b7+(b7^2)))),b8->(1/2)-((a7 (b5^2))/(2 (a7+b7)))}//Factor)==(((a7^2) solpra1)/(3 ((((b5^2)+b7 (a7+b7)))^2)))//Simplify

(eq1b/.{a1->0,b1->(((Sqrt[3]) b7 (a7+b7))/(a7 b5)),a2->0,b2->0,a3->-(((Sqrt[3]) ((b5^2)+b7 (a7+b7)))/(2 (b5^2))),b3->(((Sqrt[3]) (a7+b7) ((b5^2)-(b7^2)))/(2 a7 (b5^2))),b4->0,a4->0,a5->0,b6->0,a6->0,a8->(1/2)-(((a7^2) (b5^2))/(2 ((b5^2)+a7 b7+(b7^2)))),b8->(1/2)-((a7 (b5^2))/(2 (a7+b7)))}//Factor)==(solpra1/(3 (((a7+b7))^2)))//Simplify

Cette solution émarge du cas plus général eq171na ou sol478nou mais pas de sol478na

(eq171na/.{b4->0,a4->0,a5->0,b6->0,a6->0})==(a7^5) (b5^2) solpra1//Simplify

(sol478na/.{b4->0,a4->0,a5->0,b6->0,a6->0})//Simplify

(sol478nou/.{b4->0,a4->0,a5->0,b6->0,a6->0})==(a7^3) b7 solpra1//Simplify