Paraboles tangentes 
une définie par trois points, on a déjà calculé les coefficients a1, b1, c1, 
l'autre deux points P et Q et le point de contact M.

On retravaille ici la solution proposée par wxMAxima, non directement exploitable.

On pose :

su=sqrt(y(P)-a1*x(P)^2-b1*x(P)-c1)

sv=sqrt(y(Q)-a1*x(Q)^2-b1*x(Q)-c1)

suv=sqrt((y(P)-a1*x(P)^2-b1*x(P)-c1)*(y(Q)-a1*x(Q)^2-b1*x(Q)-c1))

1. Calcul de a3 - OK bien interprété

(2*sqrt((y(P)-a1*x(P)^2-b1*x(P)-c1)*y(Q)+(-a1*y(P)+a1^2*x(P)^2+a1*b1*x(P)+a1*c1)*x(Q)^2+(-b1*y(P)+a1*b1*x(P)^2+b1^2*x(P)+b1*c1)*x(Q)-c1*y(P)+a1*c1*x(P)^2+b1*c1*x(P)+c1^2)+y(Q)+(-2*a1*x(P)-b1)*x(Q)+y(P)-b1*x(P)-2*c1)/(x(Q)^2-2*x(P)*x(Q)+x(P)^2)

2. Calcul de b3 - à réécrire les radicandes de su et sv sont négatifs mais se factorise

-((2*su*x(Q)+2*x(P)*su)*sv+2*x(P)*y(Q)+(-2*a1*x(P)-b1)*x(Q)^2+(2*y(P)-2*a1*x(P)^2-2*b1*x(P)-2*c1)*x(Q)-b1*x(P)^2-2*c1*x(P))/(x(Q)^2-2*x(P)*x(Q)+x(P)^2)

3. Calcul de c3 - à factoriser même problème

(x(P)^2*y(Q)^2+sv*((2*x(P)*su*x(Q)+2*x(P)^2*su)*y(Q)-2*a1*x(P)*su*x(Q)^3+su*(2*y(P)-4*a1*x(P)^2-4*b1*x(P))*x(Q)^2+su*(2*x(P)*y(P)-2*a1*x(P)^3-4*b1*x(P)^2-8*c1*x(P))*x(Q))+((y(P)-3*a1*x(P)^2-b1*x(P))*x(Q)^2+(4*x(P)*y(P)-4*a1*x(P)^3-6*b1*x(P)^2-6*c1*x(P))*x(Q)+x(P)^2*y(P)-a1*x(P)^4-b1*x(P)^3-2*c1*x(P)^2)*y(Q)+(-a1*y(P)+2*a1^2*x(P)^2+a1*b1*x(P))*x(Q)^4+((-4*a1*x(P)-b1)*y(P)+4*a1^2*x(P)^3+7*a1*b1*x(P)^2+(6*a1*c1+b1^2)*x(P))*x(Q)^3+(y(P)^2+(-3*a1*x(P)^2-6*b1*x(P)-2*c1)*y(P)+2*a1^2*x(P)^4+7*a1*b1*x(P)^3+(4*a1*c1+6*b1^2)*x(P)^2+8*b1*c1*x(P))*x(Q)^2+((-b1*x(P)^2-6*c1*x(P))*y(P)+a1*b1*x(P)^4+(6*a1*c1+b1^2)*x(P)^3+8*b1*c1*x(P)^2+8*c1^2*x(P))*x(Q))/((2*su*x(Q)^2-4*x(P)*su*x(Q)+2*x(P)^2*su)*sv+(x(Q)^2-2*x(P)*x(Q)+x(P)^2)*y(Q)-a1*x(Q)^4+(2*a1*x(P)-b1)*x(Q)^3+(y(P)-2*a1*x(P)^2+b1*x(P)-2*c1)*x(Q)^2+(-2*x(P)*y(P)+2*a1*x(P)^3+b1*x(P)^2+4*c1*x(P))*x(Q)+x(P)^2*y(P)-a1*x(P)^4-b1*x(P)^3-2*c1*x(P)^2)


4. Réécriture de b3

-(2*(x(Q)+x(P))*suv+2*x(P)*y(Q)+(-2*a1*x(P)-b1)*x(Q)^2+(2*y(P)-2*a1*x(P)^2-2*b1*x(P)-2*c1)*x(Q)-b1*x(P)^2-2*c1*x(P))/(x(Q)^2-2*x(P)*x(Q)+x(P)^2)


5. Factorisation du numérateur de c3

x(P)^2*y(Q)^2+((y(P)-3*a1*x(P)^2-b1*x(P))*x(Q)^2+(4*x(P)*y(P)-4*a1*x(P)^3-6*b1*x(P)^2+(2*su*sv-6*c1)*x(P))*x(Q)+x(P)^2*y(P)-a1*x(P)^4-b1*x(P)^3+(2*su*sv-2*c1)*x(P)^2)*y(Q)+(-a1*y(P)+2*a1^2*x(P)^2+a1*b1*x(P))*x(Q)^4+((-4*a1*x(P)-b1)*y(P)+4*a1^2*x(P)^3+7*a1*b1*x(P)^2+(-2*a1*su*sv+6*a1*c1+b1^2)*x(P))*x(Q)^3+(y(P)^2+(-3*a1*x(P)^2-6*b1*x(P)+2*su*sv-2*c1)*y(P)+2*a1^2*x(P)^4+7*a1*b1*x(P)^3+(-4*a1*su*sv+4*a1*c1+6*b1^2)*x(P)^2+(8*b1*c1-4*b1*su*sv)*x(P))*x(Q)^2+(((2*su*sv-6*c1)*x(P)-b1*x(P)^2)*y(P)+a1*b1*x(P)^4+(-2*a1*su*sv+6*a1*c1+b1^2)*x(P)^3+(8*b1*c1-4*b1*su*sv)*x(P)^2+(8*c1^2-8*c1*su*sv)*x(P))*x(Q)

6. Factorisation du dénominateur de c3

(x(Q)^2-2*x(P)*x(Q)+x(P)^2)*y(Q)-a1*x(Q)^4+(2*a1*x(P)-b1)*x(Q)^3+(y(P)-2*a1*x(P)^2+b1*x(P)+2*su*sv-2*c1)*x(Q)^2+
(-2*x(P)*y(P)+2*a1*x(P)^3+b1*x(P)^2+(4*c1-4*su*sv)*x(P))*x(Q)+x(P)^2*y(P)-a1*x(P)^4-b1*x(P)^3+(2*su*sv-2*c1)*x(P)^2

7. Réécriture de c3 (en remplaçant su*sv par suv)

7.a. Numérateur

x(P)^2*y(Q)^2+((y(P)-3*a1*x(P)^2-b1*x(P))*x(Q)^2+(4*x(P)*y(P)-4*a1*x(P)^3-6*b1*x(P)^2+(2*suv-6*c1)*x(P))*x(Q)+x(P)^2*y(P)-a1*x(P)^4-b1*x(P)^3+(2*suv-2*c1)*x(P)^2)*y(Q)+(-a1*y(P)+2*a1^2*x(P)^2+a1*b1*x(P))*x(Q)^4+((-4*a1*x(P)-b1)*y(P)+4*a1^2*x(P)^3+7*a1*b1*x(P)^2+(-2*a1*suv+6*a1*c1+b1^2)*x(P))*x(Q)^3+(y(P)^2+(-3*a1*x(P)^2-6*b1*x(P)+2*suv-2*c1)*y(P)+2*a1^2*x(P)^4+7*a1*b1*x(P)^3+(-4*a1*suv+4*a1*c1+6*b1^2)*x(P)^2+(8*b1*c1-4*b1*suv)*x(P))*x(Q)^2+(((2*suv-6*c1)*x(P)-b1*x(P)^2)*y(P)+a1*b1*x(P)^4+(-2*a1*suv+6*a1*c1+b1^2)*x(P)^3+(8*b1*c1-4*b1*suv)*x(P)^2+(8*c1^2-8*c1*suv)*x(P))*x(Q)

7.b. Denominateur

(x(Q)^2-2*x(P)*x(Q)+x(P)^2)*y(Q)-a1*x(Q)^4+(2*a1*x(P)-b1)*x(Q)^3+(y(P)-2*a1*x(P)^2+b1*x(P)+2*suv-2*c1)*x(Q)^2+
(-2*x(P)*y(P)+2*a1*x(P)^3+b1*x(P)^2+(4*c1-4*suv)*x(P))*x(Q)+x(P)^2*y(P)-a1*x(P)^4-b1*x(P)^3+(2*suv-2*c1)*x(P)^2


7.c. Valeur finale de c3

(x(P)^2*y(Q)^2+((y(P)-3*a1*x(P)^2-b1*x(P))*x(Q)^2+(4*x(P)*y(P)-4*a1*x(P)^3-6*b1*x(P)^2+(2*suv-6*c1)*x(P))*x(Q)+x(P)^2*y(P)-a1*x(P)^4-b1*x(P)^3+(2*suv-2*c1)*x(P)^2)*y(Q)+(-a1*y(P)+2*a1^2*x(P)^2+a1*b1*x(P))*x(Q)^4+((-4*a1*x(P)-b1)*y(P)+4*a1^2*x(P)^3+7*a1*b1*x(P)^2+(-2*a1*suv+6*a1*c1+b1^2)*x(P))*x(Q)^3+(y(P)^2+(-3*a1*x(P)^2-6*b1*x(P)+2*suv-2*c1)*y(P)+2*a1^2*x(P)^4+7*a1*b1*x(P)^3+(-4*a1*suv+4*a1*c1+6*b1^2)*x(P)^2+(8*b1*c1-4*b1*suv)*x(P))*x(Q)^2+(((2*suv-6*c1)*x(P)-b1*x(P)^2)*y(P)+a1*b1*x(P)^4+(-2*a1*suv+6*a1*c1+b1^2)*x(P)^3+(8*b1*c1-4*b1*suv)*x(P)^2+(8*c1^2-8*c1*suv)*x(P))*x(Q))/((x(Q)^2-2*x(P)*x(Q)+x(P)^2)*y(Q)-a1*x(Q)^4+(2*a1*x(P)-b1)*x(Q)^3+(y(P)-2*a1*x(P)^2+b1*x(P)+2*suv-2*c1)*x(Q)^2+(-2*x(P)*y(P)+2*a1*x(P)^3+b1*x(P)^2+(4*c1-4*suv)*x(P))*x(Q)+x(P)^2*y(P)-a1*x(P)^4-b1*x(P)^3+(2*suv-2*c1)*x(P)^2)