Thanks a lot.

How and where to add to this code ;

triangle(ABC,[A,B,C]):
ABC(color=orange,filled=true,transparency=0.9),:

On considère les triangles inscrits dans une ellipse et tels que leur centre de gravité soit au centre de l'ellipse. Démontrer que les normales aux 3 sommets de ce triangle sont concourantes et trouver le lieu de leur point de rencontre.
restart;
with(plots);
with(geometry);
_EnvHorizontalName := 'x';
_EnvVerticalName := 'y';
HC := HorizontalCoord;
VC := VerticalCoord;
a := 11;
b := 7;
t := (3*Pi)/8;
c := sqrt(a^2 - b^2);
ellipse(e1, x^2/a^2 + y^2/b^2 = 1);
point(Oo, 0, 0);
point(A, a*cos(t), b*sin(t));
point(B, a*cos(t + 2/3*Pi), b*sin(t + 2/3*Pi));
point(C, a*cos(t + 4/3*Pi), b*sin(t + 4/3*Pi));
point(G, (A[1] + B[1] + C[1])/3, (A[2] + B[2] + C[2])/3);
line(NorA, y - A[2] = a^2*A[2]*(x - A[1])/b^2, [x, y]);
line(NorB, y - B[2] = a^2*B[2]*(x - B[1])/b^2, [x, y]);
line(NorC, y - C[2] = a^2*C[2]*(x - C[1])/b^2, [x, y]);
lieu := a^2*x^2 + b^2*y^2 - c^4/4 = 0;
Lieu := implicitplot(lieu, x = -a .. a, y = -b .. b, color = green);
tx := textplot([[coordinates(A)[], "A"], [coordinates(B)[], "B"], [coordinates(C)[], "C"], [coordinates(Oo)[], "O"]], font = [times, bold, 16], align = [above, left]);
dr := draw([e1(color = blue), NorA(color = red), NorB(color = red), NorC(color = red), A(color = red, symbol = solidcircle, symbolsize = 12), B(color = red, symbol = solidcircle, symbolsize = 12), C(color = red, symbol = solidcircle, symbolsize = 12), Oo(color = red, symbol = solidcircle, symbolsize = 12)]);
display([dr, tx, Lieu], scaling = constrained, axes = normal, title = "Ellipse et normales ", titlefont = [HELVETICA, 14]);
Warning, data could not be converted to float Matrix
Warning, data could not be converted to float Matrix
Warning, data could not be converted to float Matrix
 

What is the method for determining H1 by sol[1] or sol[2] in a similar manner to H2?

How to animate hyperbolas on a drawing to show that X0 is fixed? Thank you.

Bravo. Best Regards.
How to show that the fixed point of the line A1B1 is the projection of O on the line AB.

Oui O est l'origine (0,0).

Comment faire pour N1 et N2 fassent un tour d'ellipse complet. Merci.

Comment calculer simpliment les coordonnées de F puisque (Q,F,P,M) est une division harmonique. et comment faire trourner de segment N1N2 pour éviter la confusion des points N1 et N2 dans la rotation. Salutations empressées.

Je n'ai jamais considéré ceux qui répondaient à mes questions comme des "bonnes poires";
2 remarques :
1.- le code m'a été proposé par IA mistal qu'il m'a fallu adapter.
2.- je suis un vieux de 85 ans defficient visuel,..., et je tente de faire des mathématiques pour le plaisir.
Sincères salutations; Michel Jamet.

Pas si simple de prévoir un programme qui convienne à tous les types de coniques. Merci à tous les intervenats

I have just realized that a matrix is singular that are determinant is null and that the calculation of xc and yc is impossible

restart;
unprotect(D);
f := (x, y) -> 9*x^2 - 24*y*x + 16*y^2 + 10*x - 70*y + 175;
coeffs(f(x, y));
A, B, C, D, E, F := %;
theta := 1/2*arctan(B/(A - C));
solve({-2*A*xc - B*yc = D, -B*xc - 2*C*yc = E});
assign(%);
x := xcan*cos(theta) - ycan*sin(theta) + xc;
y := xcan*sin(theta) + ycan*cos(theta) + yc;
Eq := simplify(expand(f(x, y)));
xcan^2/simplify(sqrt(-tcoeff(Eq)/coeff(Eq, xcan^2)))^`2` + ycan^2/simplify(sqrt(-tcoeff(Eq)/coeff(Eq, ycan^2)))^`2` = 1;
Error, invalid left hand side in assignment
Error, numeric exception: division by zero
How to correct these errors ? Thank you.

sorry question misplaced; I did not confuse f1 and f2I which must be replaced by f
the code works for   f := (x, y) -> 3*x^2 - 3*y*x + 6*y^2 - 6*x + 7*y - 9 and don't work for    
                                f := (x, y) -> 9*x^2 - 24*y*x + 16*y^2 + 10*x - 70*y + 175

Based on your advice, this is what I get :
 

intersections(X^2 + Y^2 - 1, X - Y, X);
                          R￩sultant :

                               2    
                            2 Y  - 1

X=-.7071067812,   Y=-.7071067812   --->  0.\\nX=-.7071067812,   Y=.7071067812   --->  1.414213562\\nX=.7071067812,   Y=-.7071067812   --->  1.414213562\\nX=.7071067812,   Y=.7071067812   --->  0.\\nNombre de solutions :  2\n
 {[-0.7071067812, -0.7071067812], [0.7071067812, 0.7071067812]}
This is not satisfactory.
I would like a single X= and a single Y= on a line without n; Thank you

Sorry idon't understand your answer.
 

intersections(X^2 + Y^2 - 1, X - Y, X);              

X=-.7071067812,   Y=-.7071067812   --->  0.\n X=-.7071067812,   Y=.7071067812   --->  1.414213562\n X=.7071067812,   Y=-.7071067812   --->  1.414213562\n X=.7071067812,   Y=.7071067812   --->  0.\n Nombre de solutions :  2\n
 {[-0.7071067812, -0.7071067812], [0.7071067812, 0.7071067812]}
How to arrange X and Y on a line without \n; Thank you.