our program ll2 works very well; Thank a lot. How to find the equation of the geometrical place of T when the points M1 and M2 turn 

Fig:=proc(t)  global a,b,c,F1,F2,el:  local  L4,M1,M2,t1,t2,tang1,tang2,T:  t1:=t:  t2:=t+1/(2)*Pi:   point(M1, a*cos(t1), b*sin(t1)):    point(M2, a*cos(t2), b*sin(t2)):    line( tang1, x*a*cos(t1)/a^2 + y*b*sin(t1)/b^2 = 1):    line( tang2, x*a*cos(t2)/a^2 + y*b*sin(t2)/b^2 = 1):    intersection(T,tang1,tang2):  line(L4, [F2, M1]):  circle( c1, [T, distance(T, L4) ] ):    display( [ textplot               ( [ [ -c, 0, "F1"],                   [ c,  0, "F2"] ,                                            [ coordinates(M1)[], "M1"],                                             [ coordinates(M2)[], "M2"],                   [ coordinates(T)[], "T"]                 ],                 align={"above",'right'}               ),               draw               ( [ c1(color=blue),                   el(color=red),                   M1(color=black, symbol=solidcircle, symbolsize=16),                   M2(color=black, symbol=solidcircle, symbolsize=16),                   T(color=black, symbol=solidcircle, symbolsize=16),                                                 L4(color=green),                   tang1(color=blue),                   tang2(color=blue),                   F1(color=blue, symbol=solidcircle, symbolsize=16),                                       F2(color=red, symbol=solidcircle, symbolsize=16)                 ]              )            ],            scaling=constrained,            axes=none         )  end:  #` for instance `  Fig:=proc(t)  global a,b,c,F1,F2,el:  local  L4,M1,M2,t1,t2,tang1,tang2,T:  t1:=t:  t2:=t+1/(2)*Pi:   point(M1, a*cos(t1), b*sin(t1)):    point(M2, a*cos(t2), b*sin(t2)):    line( tang1, x*a*cos(t1)/a^2 + y*b*sin(t1)/b^2 = 1):    line( tang2, x*a*cos(t2)/a^2 + y*b*sin(t2)/b^2 = 1):    intersection(T,tang1,tang2):  line(L4, [F2, M1]):  circle( c1, [T, distance(T, L4) ] ):    display( [ textplot               ( [ [ -c, 0, "F1"],                   [ c,  0, "F2"] ,                                            [ coordinates(M1)[], "M1"],                                             [ coordinates(M2)[], "M2"],                   [ coordinates(T)[], "T"]                 ],                 align={"above",'right'}               ),               draw               ( [ c1(color=blue),                   el(color=red),                   M1(color=black, symbol=solidcircle, symbolsize=16),                   M2(color=black, symbol=solidcircle, symbolsize=16),                   T(color=black, symbol=solidcircle, symbolsize=16),                                                 L4(color=green),                   tang1(color=blue),                   tang2(color=blue),                   F1(color=blue, symbol=solidcircle, symbolsize=16),                                       F2(color=red, symbol=solidcircle, symbolsize=16)                 ]              )            ],            scaling=constrained,            axes=none         )    ;end:
debug(Fig);
                              Fig

Fig(Pi/4);
{--> enter Fig, args = (1/4)*Pi
                                 1   
                           t1 := - Pi
                                 4   

                                 3   
                           t2 := - Pi
                                 4   

                               M1

                               M2

                             tang1

                             tang2

                               T

                               L4

                               c1

<-- ERROR in Fig (now at top level) = expecting plot structures but received: %1, [plots:-textplot*([[-2*10^(1/2), 0, F1], [2*10^(1/2), 0, F2], [(11/2)*2^(1/2), (9/2)*2^(1/2), M1], [-(11/2)*2^(1/2), (9/2)*2^(1/2), M2], [0, 9*2^(1/2), T]], align = {above, right}), geometry:-draw*[c1(color = blue), el(color = red), M1(color = black, symbol = solidcircle, symbolsize = 16), M2(color = black, symbol = solidcircle, symbolsize = 16), T(color = black, symbol = solidcircle, symbolsize = 16), L4(color = green), tang1(color = blue), tang2(color = blue), F1(color = blue, symbol = solidcircle, symbolsize = 16), F2(color = red, symbol = solidcircle, symbolsize = 16)]]}
Error, (in plots:-display) expecting plot structures but received: [plots:-textplot*([[-2*10^(1/2), 0, "F1"], [2*10^(1/2), 0, "F2"], [(11/2)*2^(1/2), (9/2)*2^(1/2), "M1"], [-(11/2)*2^(1/2), (9/2)*2^(1/2), "M2"], [0, 9*2^(1/2), "T"]], align = {"above", right}), geometry:-draw*[c1(color = blue), el(color = red), M1(color = black, symbol = solidcircle, symbolsize = 16), M2(color = black, symbol = solidcircle, symbolsize = 16), T(color = black, symbol = solidcircle, symbolsize = 16), L4(color = green), tang1(color = blue), tang2(color = blue), F1(color = blue, symbol = solidcircle, symbolsize = 16), F2(color = red, symbol = solidcircle, symbolsize = 16)]]
nFig := 60.0;
Figs := seq(Fig1(2*Pi*i/nFig), i = 10 .. nFig - 1.0);
display(Figs, insequence = true);
                          nFig := 60.0

#Corona--45-p9
#Les 4 rayons-vecteurs qui joignent les foyers d'une ellipse à 2 pointsM, M' de la courbe sont tangents à même cercle
restart;
with(plots):
with(geometry):
_EnvHorizontalName := 'x':
_EnvVerticalName := 'y':

a := 11;
b := 9:
c := sqrt(a^2 - b^2);
t1 := (3*Pi)/4;
t2 := (-4*Pi)/5;
ellipse(el, x^2/a^2 + y^2/b^2 - 1);
point(M1, a*cos(t1), b*sin(t1)):
point(M2, a*cos(t2), b*sin(t2)):
point(F1, -c, 0);
point(F2, c, 0);
line(L1, [F1, M1]):
line(L2, [F2, M2]):
line(L3, [F1, M2]):
line(L4, [F2, M1]):
line( tang1,x*a*cos(t1)/a^2 + y*b*sin(t1)/b^2 = 1):
line( tang2,x*a*cos(t2)/a^2 + y*b*sin(t2)/b^2 = 1): 
intersection(T,tang1,tang2):

distance(T, L4):r:=evalf(%):

                            a := 11

                                  (1/2)
                         c := 2 10     

                                 3   
                           t1 := - Pi
                                 4   

                                  4   
                          t2 := - - Pi
                                  5   

                               el

                               F1

                               F2

with(plottools):

c1 := circle(coordinates(T), r, color = blue):
display((c1),
  ( [ textplot
      ( [ [ -c, 0, "F1"],
          [ c,  0, "F2"] ,                         
          [ coordinates(M1)[], "M1"],                          
          [ coordinates(M2)[], "M2"],
          [ coordinates(T)[], "T"]                           
          
        ],
        align={"above",'right'}
      ),
      draw
      ( [ el(color=red),
          M1(color=black,symbol=solidcircle, symbolsize=16),
          M2(color=black,symbol=solidcircle, symbolsize=16),
          T(color=black,symbol=solidcircle, symbolsize=16),            
          L1( color=black),
          L2(color=green),
          L3( color=black),
          L4(color=green),
          tang1(color=blue), 
          tang2(color=blue),
          F1(color=blue, symbol=solidcircle, symbolsize=16),                      
          F2(color=red, symbol=solidcircle, symbolsize=16)
          
        ],
        axes=none
      )
    ]
  );

Error, `;` unexpected 
it is very difficult to find this error; Thank you for your help.

 Fig := proc(t)
              local M1,M2,m,L1,L3,L4,tang1,tang2,C1,P,T1,xM2,yM2;
              global a, b, c,el,F1,F2,OO,cir;
              point(M1, a*cos(t), b*sin(t)); 
              line( L1, [F1, M1]): 
              EQ([a*cos(t),b*sin(t)],[-c,0]):  
              solve(`%`,y):
              m:=coeff(`%`,x):   
              line(L3,y=m*(x-c)): 
              op(solve({x^2/a^2 + y^2/b^2 - 1 =0, y=m*(x-c)},{x,y},explicit)[2])[1]:
              xM2:=rhs(`%`):  
              op(solve({x^2/a^2 + y^2/b^2 - 1 =0, y=m*(x-c)},{x,y},explicit)[2])[2]:
              yM2:=rhs(`%`):  
              point(M2,xM2,yM2): 
              line( tang1,x*xM2/a^2 + y*yM2/b^2 = 1):  
              line( tang2,x*cos(t)/a + y*sin(t)/b = 1): 
              intersection(C1,L1,tang1): 
              midpoint(P,M2,C1): 
             #intersection(P,tang1,tang2): 
              line(L4,[F1,M2]):
              triangle(T1, [C1,F1,M2]): 
 display
  ( [ textplot
      ( [ [ -c, 0, "F1"],
          [ c,  0, "F2"] ,
          [ coordinates(P)[], "P"],                          
          [ coordinates(M1)[], "M1"],                          
          [ coordinates(M2)[], "M2"],                         
          [ coordinates(C1)[], "C1"]
        ],
        align={"above",'right'}
      ),
      draw
      ( [ el(color=red),
          M1(color=black,symbol=solidcircle, symbolsize=16),
          M2(color=black,symbol=solidcircle, symbolsize=16),                      
          L1( color=black),
          L3(color=green),
          L4(color=green),                      
          tang1(color=red),  tang2(color=red),cir(color=black),
          P(color=blue, symbol=solidcircle, symbolsize=16),   
          F1(color=blue, symbol=solidcircle, symbolsize=16),                      
          F2(color=red, symbol=solidcircle, symbolsize=16),
          C1(color=black, symbol=solidcircle, symbolsize=16),
          T1(color=blue, filled=true, transparency=0.95)
        ],
        axes=none
      )
    ]
  );
end:
      
  
 nFig := 180.0:

   Figs := seq(Fig(2*Pi*i/nFig), i = 0 .. nFig-1.0):
   display(Figs, insequence = true);
I eliminated an intersection but there is still a problem with P, How to solve it. Thank you.

 Fig := proc(t)
              local M1,M2,m,L1,L3,L4,tang1,tang2,C1,P,T1,xM2,yM2;
              global a, b, c,el,F1,F2,OO,cir;
              point(M1, a*cos(t), b*sin(t)); 
              line( L1, [F1, M1]): 
              EQ([a*cos(t),b*sin(t)],[-c,0]):  
              solve(`%`,y):
              m:=coeff(`%`,x):   
              line(L3,y=m*(x-c)): 
              op(solve({x^2/a^2 + y^2/b^2 - 1 =0, y=m*(x-c)},{x,y},explicit)[2])[1]:
              xM2:=rhs(`%`):  
              op(solve({x^2/a^2 + y^2/b^2 - 1 =0, y=m*(x-c)},{x,y},explicit)[2])[2]:
              yM2:=rhs(`%`):  
              point(M2,xM2,yM2): 
              line( tang1,x*xM2/a^2 + y*yM2/b^2 = 1):  
              line( tang2,x*cos(t)/a + y*sin(t)/b = 1): 
              intersection(C1,L1,tang1): 
              intersection(P,tang1,tang2): line(L4,[F1,M2]):
              triangle(T1, [C1,F1,M2]): 
      draw
      ( [ el(color=red),
          M1(color=black,symbol=solidcircle, symbolsize=16),
          M2(color=black,symbol=solidcircle, symbolsize=16),                      
          L1( color=black),
          L3(color=green),
          L4(color=green),                      
          tang1(color=red),  tang2(color=red),cir(color=black),
          P(color=blue, symbol=solidcircle, symbolsize=16),   
          F1(color=blue, symbol=solidcircle, symbolsize=16),                      
          F2(color=red, symbol=solidcircle, symbolsize=16),
          C1(color=black, symbol=solidcircle, symbolsize=16),
          T1(color=blue, filled=true, transparency=0.95)
        ],
        axes=none
      ):end:
  
 nFig := 180.0:

   Figs := seq(Fig(2*Pi*i/nFig), i = 0.1 .. nFig-1.0):
   display(Figs, insequence = true);
intersection: two given lines are ParallelLine, no intersection
Error, (in geometry:-draw) unknown geometric object  P
Error, (in plots:-display) expecting plot structure but received: Figs
How to overcome this difficulty? Thank a lot.

I am sorry; The program does not work With messages : 
Error, (in geometry:-draw) improper op or subscript selector
Error, (in plots:-display) expecting plot structure but received: Figs

Is there any way how to suppress the q on p jumps ? Thank you.

Fig := proc(t) local M, l1, q; global a, b, p; point(M, a*cos(t), b*sin(t)); line(l1, y = (Student[Calculus1]):-Tangent(solve(Equation(p), y)[2], x = HorizontalCoord(M))); reflection(q, p, l1); draw([p(color = blue), q(color = blue), l1(color = black), M(color = red, symbol = solidcircle, symbolsize = 16)], axes = normal, scaling = constrained); end proc
nFig := 60;
Figs := seq(Fig(2*Pi*i/nFig), i = 1 .. nFig);
Error, (in Student:-Calculus1:-Tangent) the slope is not defined at the point `x` = -7
display(Figs, insequence = true);
Error, (in plots:-display) expecting plot structure but received: Figs

Iam sorry; How to manage this error. Thank you.
 

I want to trace the tangent in P to p 

a := 7;
b := 3;
t := (2*Pi)/3;
NULL;
ellipse(p, x^2/a^2 + y^2/b^2 - 1, [x, y]);
point(P, a*cos(t), b*sin(t));
NULL;
rotation(q, p, t, 'clockwise', P);
tangentpc(Tg, P, p);
Error, (in geometry:-tangentpc) wrong type of arguments

draw([p(color = blue), P, q(color = blue),Tg(color=red), axes = normal, scaling = constrained);
Why this error message.? Thank you.

how to highlight that the sum of the sides is constant ? Thank you.
 Fig := proc(t) 
              local xD,yD, D, C,Points,tex; 
              global A, B, b, Omega1, EL1, EL2;
              xD:=Omega1[1]+aa*cos(t):
              yD:=bb*sin(t):
              D:= [xD, yD]; C := [xD + b, yD];
              Points := pointplot([A[], B[], C[], C[], D[], E[], Omega1[]], symbol = solidcircle, color = [red],     symbolsize = 6): 
              tex := textplot([[A[], "A"], [B[], "B"], [C[], "C"], [D[], "D"], [E[], "E"], [Omega1[], "&Omega;1"]], align = ["above", "right"]):
              display([polygonplot([A, B, C, D], color = blue, filled = true, transparency = 0.9), Points,tex,EL1, EL2], axes = normal, scaling = constrained); 
         end proc:
Fig((3*Pi)/4);
display([seq(Fig((2*Pi*i)/40), i = 1 .. 80)], insequence = true):
 

Thank you for your answer , And without geometry package ?

I still have a problem
 restart:
  with(plots):
  with(plottools):
  _EnvHorizontalName:= 'x':
  _EnvVerticalName:= 'y':
  EqBIS:= proc(P, U, V) 
               local a, eq, eq1, M1, t, PU, PV, bissec1:
               uses geometry, LinearAlgebra:
               description "P est le sommet de l'angle dont on chercche la bissectrice" :
               a:= (P - U)/Norm(P - U, 2) + (P - V)/Norm(P - V, 2): 
               M1:= P + a*t:
               eq1:= op(eliminate({x = M1[1], y = M1[2]}, t)): 
               eq:= op(eq1[2]):
               line(bis, eq):
               line(lv, x = 2*sqrt(r1*R)):
               intersection(Omega, bis, lv):
               return coordinates(Omega);
         end proc:
  r1:= 1/2:
  r2:= r1/2:
  R:= r1*(21 - 12*sqrt(3)):
  R:= r1*(21 - 12*sqrt(3)):
  a:= arc([0, 0], 2*r1, Pi/6 .. (5*Pi)/6):
  b:= arc([0, 0], r1, Pi/6 .. (5*Pi)/6):
  co:= EqBIS(<sqrt(3)/2, 1/2>, <0, 0>, <0, 1/2>):
  A:=[cos(Pi/6), sin(Pi/6)]:
  B:=[cos(5*Pi/6), sin(5*Pi/6)]:
  Oo:=[0,0]:
  Op:=[0,1/2]:
  poly:=[A,B,Oo]:
  Omega:= [co[1], co[2]]:
  Omega1:= [-co[1], co[2]]:
  r3:= 3/16:
  EF:= sqrt(r3):
  r:= (150 - 72*sqrt(3))/193*1/2:
  alpha:= -5/3*r + 1/2*1/2:
  p:= sqrt(3)/3*1/2 - sqrt(3)/18*r:
  p2:= textplot([[A[], "A"], [B[], "B"], [Oo[], "O"]], align = ["above", "right"]):
  display( [ a,
             b,
             seq(plot([[1/2*cos(Pi/6+i*1/5), 1/2*sin(Pi/6+i*1/5)],[0,0]],i=1..10),color=black),
             sector([0,0], 1/2, Pi/6.. 5*Pi/6, color =white,transparency=0.3),
             polygonplot(poly, thickness = 3, color = blue,transparency=0.7),
             disk(Omega, R, color = blue),
             disk(Omega1, R, color = blue),
             disk([0, 3/4], 1/4, color = yellow),
             disk([EF, 1/2 + r3], r3, color = green), disk([-EF, 1/2 + r3], r3, color = green),
             circle([-EF, 1/2 + r3], r3, color = black),
             circle([p, 3/4 + alpha], r, color = black),disk([p, 3/4 + alpha], r, color = red),
             circle([-p, 3/4 + alpha], r, color = black),disk([-p, 3/4 + alpha], r, color = red)
          ],
          axes = none,
          scaling = constrained,
          size = [500, 500]
        );
Warning, data could not be converted to float Matrix, I don't understand this message. Thank you for your help.
 

Thank you for your answer; I found : 

restart;
with(plots):
with(geometry):
_EnvHorizontalName := x:
_EnvVerticalName := y:
R := 11:
r := 7:
a := sqrt(R*r):

b := 2:
circle(C1, [point(P1, [0, 0]), R]):
circle(C2, [point(P2, [R + 2*b + r, 0]), r]):
ellipse(p, (x - R - b)^2/b^2 + y^2/a^2 = 1):
draw([C1(color = yellow, filled = true), 
C2(color = red, filled = true), p(color = blue, filled = true), 
C1(color = black), C2(color = black), p(color = black)], 
axes = none, view = [-15 .. 35, -15 .. 15], scaling = constrained):
alpha := arctan((R - r)/(R + 2*b + r));
long := cos(alpha)*(R + 2*b + r);
evalf(%);
circle(C2, [point(P2, [long, r - R]), r]);
rotation(p1, p, alpha, 'clockwise');
detail(p1);
point(A, 0, -R);
point(B, long, -R);
line(L1, [A, B]);
point(cen, [(143*sqrt(5))/25, -(26*sqrt(5))/25]);
line(L, [P1, P2]);
reflection(L2, L1, L);
detail(L2);
draw([C1(color = yellow, filled = true), C2(color = red, filled = true), p1(color = blue, filled = true), C1(color = black), C2(color = black), p1(color = black), L1(color = black), L2(color = black)], axes = none, view = [-15 .. 35, -15 .. 15], scaling = constrained);
The drawing is a sangatku.

restart;
with(plots):
with(geometry):
_EnvHorizontalName := x:
_EnvVerticalName := y:
R := 7:
point(A, [0, R]):
line(L1, y = sqrt(3)*x + R):
line(L2, y = -sqrt(3)*x + R):
line(L3, y = R/3):
intersection(B, L1, L3):
intersection(C, L2, L3):

triangle(ABC, [A, B, C]):
circle(C1, [point(P1, [0, 0]), R]):
circle(C2, [point(P2, [0, R/3 + (2*R)/9]), (2*R)/9]):

center(C2), coordinates(center(C2)):
circle(C3, [reflection(P3, center(C2), C), radius(C2)]):
circle(C4, [reflection(P4, center(C2), B), radius(C2)]):
circle(C5,[reflection(P5, center(C3), L2), radius(C2)]):

detail(C5):
circle(C6, [point(P5, [0, R/3 - (2*R)/9]), radius(C2)]):
circle(C7, [point(P7,[0, R/3 - (2*R)/9-(4*R)/9]), radius(C2)]):
circle(C8, [point(P8,[0, R/3 - (2*R)/9-(8*R)/9]), radius(C2)]):

circle(Cir, [point(Po, [0, -R/3 ]), (2*R)/3]):
draw([L1(color = blue), 
ABC(color = red, transparency = 0.5, filled = true), 
L2(color = blue), L3(color = blue), 
C1(color = blue, thickness = 3), 
C1(color = yellow, transparency = 0.8, filled = true),
C2(color = blue, filled = true),
C3(color = blue, filled = true),
C4(color = blue, filled = true),
C5(color = blue, filled = true),
C6(color = blue, filled = true),
C7(color = blue, filled = true),
C8(color = blue, filled = true),
Cir(color = red,transparency = 0.5,filled=true)], 
axes = normal,
 view = [-R .. R, -R .. R], 
scaling = constrained);
I think a troucle with a symetry in respect to L2

I try differently
rotation(pol2,pol1,Pi/3,clockwise,o):
Error, (in geometry:-rotation) wrong type of arguments   Still that kind of error.