restart;
Fig:=proc(t)
local a,b,c,A,B,C,Oo,P,NorA,NorB,NorC,lieu,Lieu,dr,tx:
uses plots, geometry;
a := 11:b := 7:
c := sqrt(a^2 - b^2):

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(Oo,0,0):
lieu:=a^2*x^2+b^2*y^2-c^4/4=0:
Lieu := implicitplot(lieu, x = -a .. a, y = -b .. b, color = green):

line(NorA, y-coordinates(A)[2] =((a^2*coordinates(A)[2])/(b^2*coordinates(A)[1]))*(x-coordinates(A)[1]),[x, y]):
line(NorB, y-coordinates(B)[2] =((a^2*coordinates(B)[2])/(b^2*coordinates(B)[1]))*(x-coordinates(B)[1]), [x, y]):
line(NorC, y-coordinates(C)[2] =((a^2*coordinates(C)[2])/(b^2*coordinates(C)[1]))*(x-coordinates(C)[1]),[x, y]):
intersection(P,NorA,NorB):

ellipse(p, x^2/a^2 + y^2/b^2 - 1, [x, y]);

tx := textplot([[coordinates(A1)[], "A"],[coordinates(A2)[], "B"], [coordinates(C)[], "C"], [coordinates(Oo)[], "O"],#[coordinates(P)[], "P"]], font = [times, bold, 16], align = [above, left]):
dr := draw([p(color = blue),NorA(color=red,NorB(color=red),NorC(color=red),p(color=blue),
Oo(color = black, symbol = solidcircle, symbolsize = 8), P(color = black, symbol = solidcircle, symbolsize = 8)]):
display(dr,tx,Lieu,scaling=constrained, axes=none,title = "Les triangles inscrits dans une ellipse ont leur centre de gravité en son centre . Lieu du point de concours des perpendicalaires issues des sommets", titlefont = [HELVETICA, 14]);
end:

Error, `:=` unexpected
plots:-animate(Fig, [t], t=0.1..2*Pi, frames=150);
 

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);
eval(coordinates(G));
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]);
      [1        1        1       1        1        1     ]
      [- A[1] + - B[1] + - C[1], - A[2] + - B[2] + - C[2]]
      [3        3        3       3        3        3     ]

                              NorA

                              NorB

                              NorC

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

On joint un point M  d'une ellipse aux foyers F1 et F2.  Les droites MF1 et MF2 recoupent l'ellipse aux points H1 et H2 ,  trouver l'enveloppe de la droite H1H2,  quand le point M se `déplace` sur l'ellipse.;

restart;
Fig := proc(t) local a, b, c, courbe, sol, sol1, dr, tx; _EnvHorizontalName := 'x'; _EnvVerticalName := 'y'; a := 11; b := 7; c := sqrt(a^2 - b^2); geometry:-ellipse(e1, x^2/a^2 + y^2/b^2 = 1); geometry:-point(Oo, 0, 0); geometry:-point(M, a*cos(t), b*sin(t)); geometry:-point(F1, -c, 0); geometry:-point(F2, c, 0); geometry:-line(MF1, [M, F1]); geometry:-line(MF2, [M, F2]); sol := solve({geometry:-Equation(MF1), x^2/a^2 + y^2/b^2 = 1}, {x, y}, explicit); print(%); geometry:-point(H1, subs(sol[2], x), subs(sol[2], y)); geometry:-line(MH1, [M, H1]); sol := solve({geometry:-Equation(MF2), x^2/a^2 + y^2/b^2 = 1}, {x, y}, explicit); print(%); geometry:-point(H2, subs(sol[2], x), subs(sol[2], y)); geometry:-line(MH2, [M, H2]); courbe := plots:-implicitplot(x^2/a^2 + (a^2 + c^2)^2*y^2/b^2 - 1 = 0, x = -a .. a, y = -b .. b, color = cyan); tx := plots:-textplot([[geometry:-coordinates(M)[], "M"], [geometry:-coordinates(Oo)[], "O"], [geometry:-coordinates(H1)[], "H1"], [geometry:-coordinates(H2)[], "H2"], [geometry:-coordinates(F1)[], "F1"], [geometry:-coordinates(F2)[], "F2"]], font = [times, bold, 16], align = [above, left]); dr := geometry:-draw([e1(color = blue), MH1(color = magenta), MH2(color = magenta), M(color = red, symbol = solidcircle, symbolsize = 12), H1(color = red, symbol = solidcircle, symbolsize = 12), H2(color = red, symbol = solidcircle, symbolsize = 12), F1(color = red, symbol = solidcircle, symbolsize = 12), F2(color = red, symbol = solidcircle, symbolsize = 12), Oo(color = red, symbol = solidcircle, symbolsize = 12)]); plots:-display([dr, tx, courbe], scaling = constrained, axes = normal, title = "Ellipse et normales ", titlefont = [HELVETICA, 14]); end proc;
Fig(Pi/3);
    /    11      7  (1/2)\    /      26411   210177  (1/2)  
   { x = --, y = - 3      }, { x = - ----- - ------ 2     , 
    \    2       2       /    \      57074   28537          

         11319  (1/2)  (1/2)   66199  (1/2)\ 
     y = ----- 3      2      - ----- 3      }
         28537                 57074       / 

     /    11      7  (1/2)\    /    210177  (1/2)   26411  
    { x = --, y = - 3      }, { x = ------ 2      - -----, 
     \    2       2       /    \    28537           57074  

            11319  (1/2)  (1/2)   66199  (1/2)\ 
      y = - ----- 3      2      - ----- 3      }
            28537                 57074       / 

Fig(Pi/6);
   /      104027  (1/2)   17787  (1/2)  (1/2)   123420  (1/2)
  { x = - ------ 3      + ----- 3      6      - ------ 2     
   \      22226           11113                 11113        

       19404  (1/2)  (1/2)        66199   11319  (1/2)\   
     + ----- 2      6     , y = - ----- + ----- 6      }, 
       11113                      22226   11113       /   

     /    11  (1/2)      7\ 
    { x = -- 3     , y = - }
     \    2              2/ 

Error, (in geometry:-line) the line is not defined
plots:-animate(Fig, [t], t = 0.1 .. 2*Pi, frames = 150);
            {x = -10.99908244, y = -0.09041172732}, 

              {x = 10.94504582, y = 0.6988339166}

Error, (in plots/animate) the line is not defined
;
NULL;
Thank you for your help.

intersections := proc(P, Q, T)
local R, W, w, t, a, b, sol, buff, v;
sol := []; if T = Y then W := X; else W := Y; end if;
R := resultant(P, Q, T);
print(`Résultant :`);
print(R);
w := fsolve(R, W); t := [];
for v in [w] do t := [op(t), fsolve(subs(W = v, P), T)]; end do;
for a in [w] do for b in [t] do if T = Y then buff := abs(subs(X = a, Y = b, P)) + abs(subs(X = a, Y = b, Q));
printf(`X=%a,   Y=%a   --->  %a\\n`, a, b, buff); if buff < 1/100000000 then sol := [op(sol), [a, b]]; end if;
else buff := abs(subs(X = b, Y = a, P)) + abs(subs(X = b, Y = a, Q));
printf(`X=%a,   Y=%a   --->  %a\\n`, b, a, buff); if buff < 1/100000000 then sol := [op(sol), [b, a]];
end if; end if; end do; end do;
printf(`Nombre de solutions :  %a\\n`, nops(sol)); print(sol); end proc;
Try with :
intersections(X^2 + Y^2 - 1, X - Y, X); Would you like to develop this procedure which does not give the number of solutions ?Thank you.

On considère une ellipse x^2/a^2+y^2/b^2-1=0 et 2 sommets de cette ellipse A(a,0) et B(0,b). On imagine une hyperbole équilatère variable passant par les points O, A et B. Cette courbe rencontre l'ellipse en 2 autres points A1 et B1. Montrer que la droite A1B1 passe par un point fixe. Même avec l'intelligence artificielle, je ne parviens pas à résoudre ce problème. Pourriez-vous d'aider. Merci.

Machine translation by moderator:

We consider an ellipse x^2/a^2+y^2/b^2-1=0 and 2 vertices of this ellipse A(a,0) and B(0,b). We imagine a variable equilateral hyperbola passing through the points O, A and B. This curve meets the ellipse at 2 other points A1 and B1. Show that the line A1B1 passes through a fixed point. Even with artificial intelligence, I can't solve this problem. Could you help. Thank you.