Thank you a lot for your answer.
One observation : good explanations are given in internet site : WolframMathWorld Pell equation.

pellsolve := proc(D::posint)
local P, Q, a, A, B, i; if type(sqrt(D), integer) then error "D must be a nonsquare integer"; end if;
P := 0; Q := 1; a := floor(sqrt(D)); A := 1, a; B := 0, 1;
for i do P := a*Q - P;
Q := (D - P^2)/Q;
a := floor((P + sqrt(D))/Q);
A := A[2], a*A[2] + A[1]; B := B[2], a*B[2] + B[1]; break; if Q = 1 and i mod 2 = 0;
end do; return A[1], B[1]; end proc;
pellsolve(28);for example
####retourne la solution minimale (x,y) de l'équation de Pell x`^2 - Dy = 1, Fonctionne bien.

I’m surprised I can’t find the same coordinates using the barycentric coordinates !

Vdot := proc (U, V)
local i: add(U[i]*V[i], i = 1 .. 2) end:
dist := proc (M, N) sqrt(Vdot(expand(M-N), expand(M-N))) end:
NULL;
A := [0, 0]:
B := [7, 0]:
C := [3, 5]:
a := dist(B, C):
b := dist(A, C):
c := dist(A, B):
An := arccos((-a^2 + b^2 + c^2)/(2*b*c));
Bn := arccos((a^2 - b^2 + c^2)/(2*a*c));
Cn := arccos((a^2 + b^2 - c^2)/(2*a*b));
evalf(An + Bn + Cn);
(A[1]*csc(An + Pi/3) + B[1]*csc(Bn + Pi/3) + C[1]*csc(Cn + Pi/3))/(csc(An + Pi/3) + csc(Bn + Pi/3) + csc(Cn + Pi/3)), (A[2]*csc(An + Pi/3) + B[2]*csc(Bn + Pi/3) + C[2]*csc(Cn + Pi/3))/(csc(An + Pi/3) + csc(Bn + Pi/3) + csc(Cn + Pi/3));
evalf(%);
Do you see why ? Thank you.

Thanks a lot. Now how to locate vertex and focus of the parabola in the base (O,i,j) ?

Very good for the order of the coefficients but there is still a confusion between parabola and ellipse. How to avoid it ? Thank you.

Thank you very much. Without Geometry.

In fact I have the point K of coordinates are X, Y and I look for the place when m varies; Can you help me ? Thank you.

points:=seq(seq([xt[i][j], yt[i][j]], j=0..zt-1), i=1..2*n-1):
  p_cross:= pointplot([points], connect=true,scaling = constrained, color = black,linestyle=solid, filled=[yellow]):
  polygonplot([points], color = yellow, scaling = constrained):
  NULL:
  display([p_cross]); This drawing is obviously not suitable. How to amend it? Thabk you very much.

How to move the oscillating bar (B,C,P in the bar) or move point B

restart;

with(plots):

with(plottools):

R := 1: # length of OA

L := 5*R: # horizontal distance between O and C

H := 3*R: # vertical distance between O and C

a := 1.3*H: # length of AB

b := L - R:   # length of BC

A := R*<cos(t), sin(t)>:

B := A + a*<cos(s(t)), sin(s(t))>:  # s(t) is the angle of AB relative to the horizontal

C := [L, a]:

                         C := [5, 3.9]

# derive a differential equation for s(t)

B -~ C:

%^+ . % = b^2:

diff(%, t):

isolate(%, diff(s(t),t)):

de := simplify(%):

ic := s(0)=Pi/2:

dsol := dsolve({de,ic}, numeric, output=operator):

Head := translate(rotate(scale(polygonplot([[-60, 24], [63, -17], [60, -55.5], seq([84*cos(k1*Pi/100), 84*sin(k1*Pi/100)], k1 = -24 .. 7), [82, 18], [78, 20], [64, -3], [-46.5, 35]], scaling = constrained, color = "Gold"), 0.049, 0.03), 0.211, [0, 0]), 0, -0.3);

#Head := scale(polygonplot([ [-16,-1], [10, -1], [13,-5], [12,1], [-12,1]], scaling=constrained, color="Gold"), 0.4,#0.3):

Crank := scale(polygonplot([[-8, 6], [-7, 20], [-6, 38], [6, 38], [7, 20], [8, 6], [-10, 6], [-16, 6], [-25, 0], [-17, -8], [-8, -8], [-8, -17], [8, -17], [8, -8], [17, -8], [25, 0], [16, 6]], scaling = constrained, color = "Aquamarine"), 0.06, 0.04);

frame := proc(t)  # 0 <= t <= 2*Pi

local A, B, P, u;

uses plottools;

A := R*[cos(t), sin(t)];

A + [a*cos(s(t)), a*sin(s(t))];

B := eval(%, dsol);

P := 2*C - B;

C -~ B;

u := arctan(%[2],%[1]);  # angle of rotation of BC

        display(

circle([0,0], R),

plot([ [0,0], A, B, C, P ], style=pointline, symbol=solidcircle, symbolsize=20),

textplot([ [0,0,"O"], [A[],"A"], [B[],"B"], [C[],"C"], [P[],"P"] ], align={above}, font=[Times, bold, 18]),

translate(rotate(Head, u), C[1], C[2]),

                rotate(Crank,t-Pi/2),

                plot([A,B],color=red,thickness=8),

NULL, scaling=constrained);

end proc:

nframes := 60:

frames := seq(frame(2*Pi*i/nframes), i=0..nframes-1):

display(frames, insequence,size=[600,600]):

Thank you very much.

Thanks a lot for this animation. Now how to set "Head" on the skeleton ? Thank you. Head:=proc(k) local r,poly,k1,tC,tD,tE,DikC,DikD,DikE: global C,D,E: r:=84: C:=[-55,17]:D:=[0,0]:E:=[84,0]: poly:=[[-60,24],[63,-17],[60,-55.5],seq([r*cos((k*Pi)/(100)),r*sin((k*Pi)/(100))],k1=-24..7),[82,18],[78,20],[64,-3],[-46.5,35]] : tC:= textplot([C[], "C"], align={above, right}, font=[Times, bold, 18]): tD:=textplot([D[], "D"], align={above, left}, font=[Times, bold, 18]): tE:=textplot([E[], "E"], align={above, left}, font=[Times, bold, 18]): DikC:=disk(C,1,color=black): DikD:=disk(D,1,color=black): DikE:=disk(E,1,color=black): if 0<= k and k<=0.579 then display( polygon(poly,color=yellow), tC,tD,tE,DikC,DikD,DikE,scaling=constrained):rotate(`%`,k,D) elif -0.68<= k and k<=0 then display( polygon(poly,color=yellow), tC,tD,tE,DikC,DikD,DikE,scaling=constrained):rotate(`%`,k,D) fi

Bravo your rods are not elastic...Thank you.

The BC and BD rods do not retain the same length, so there is a difficult error to detect in the program.

How to shaw that any element of R² is a linear combination of (1,1) and (1,2), this decomposition is unique. The family is free and generates R². Thank you.

How to know if 2 pairs of vectors generate the same vector space ?
The 2 pairs of vectors are <1,0-1>  and <0,1,0> for une part, <2,1,-2> and <-1,2,,1> for the other. Thank you.

Sorry, I don't understand what are M, V, and sys.
How to get a base line of the linear application image ? Thank you.