how to draw these 3 lines and then project them on the plans Oxy,Oxz,Oyz;
3 given lines a := [3*t-7, -2*t+4, 3*t+4]; b := [m+1, 2*m-9, -m-12];c:={x = -200/29-2*t, y = 114/29+3*t, z = 119/29+4*t}, how to show these lines and the projections on the 3 planes ? Thank you.

A := [1, -2, 3]:u := `<,>`(0, -2, 2):v := `<,>`(5, 8, -3):
PL := proc (A, u, v) local d, Det, AP, t, U, V;
AP := `<,>`(x-A[1], y-A[2], z-A[3]);
Matrix(`<|>`(`<,>`(AP), `<,>`(u), `<,>`(v)));
Det := LinearAlgebra:-Determinant(%); d := igcd(coeff(Det, x), coeff(Det, y), coeff(Det, z), tcoeff(Det));
print(`Une équation cartésienne du plan est :`);
t := Det/d; print([t = 0]);
print('Une*représentation*paramétrique*du*plan*est; -1');
U := convert(u, list); V := convert(v, list);
[x = lambda*U[1]+mu*V[1]+A[1], y = lambda*U[2]+mu*V[2]+A[2], z = lambda*U[3]+mu*V[3]+A[3]] end proc;
PL(A, u, v);

plan3p := proc (A::list, B::list, C::list)
local d, M, N, P, Mat, Det, t, U, V;
M := `<,>`(x-A[1], B[1]-C[1], C[1]-A[1]); N := `<,>`(y-A[2], B[2]-C[2], C[2]-A[2]); P := `<,>`(z-A[3], B[3]-C[3], C[3]-A[3]);
Mat := Matrix([M, N, P]);
Det := LinearAlgebra:-Determinant(%);
d := igcd(coeff(Det, x), coeff(Det, y), coeff(Det, z), tcoeff(Det));
print(`Une équation cartésienne du plan est :`);
t := Det/d; print([t = 0]); print('Une*représentation*paramétrique*du*plan*est; -1');
U := A-B; V := B-C;
[x = lambda*U[1]+mu*V[1]+A[1], y = lambda*U[2]+mu*V[2]+A[2], z = lambda*U[3]+mu*V[3]+A[3]] end proc;
A := [-6, 3, -2]; B := [5, 2, 1]; C := [2, 5, 2];plan3p(A, B, C);
How to know if these procedures are correct or not. Thank you.

We consider a fixed circle (C) tangent to a fixed line Δ at a given point O of this line. 
Circles Γ tangent to circles C in M and to the right Δ in N are studied.
 Show that the MN line passes through a fixed point I. Infer that the circles Γ remain orthogonal to a fixed circle.
My code is :
 restart; with(geometry);with(plots);
_EnvHorizontalName := 'x';_EnvVerticalName := 'y';
dist := proc (M, N) sqrt(Vdot(expand(M-N), expand(M-N))) end proc;
point(oo, 0, 3); p := 6;
point(N, 5, 0);
line(Delta, y = 0, [x, y]);
para := x^2 = 2*p*y;
solve(subs(x = 5, para), y); point(varpi, 5, 25/12);
line(alpha, [oo, varpi]); k := 3/(25/12);
point(M, (0+5*k)/(1+k), (3+25*k*(1/12))/(1+k)); 
circle(C, x^2+(y-3)^2 = 9, [x, y]);cir := implicitplot(x^2+(y-3)^2 = 9, x = -5 .. 5, y = -5 .. 7, color = blue);
Para := implicitplot(para, x = -40 .. 40, y = 0 .. 40, linestyle = 3, color = coral);
homothety(J, N, -k, M); coordinates(J);
circle(C1, (y-25/12)^2+(x-5)^2 = (25/12)^2, [x, y]);line(lNJ, [N, J]);
triangle(T1, [J, oo, M]); triangle(T2, [N, varpi, M]);
C1 := implicitplot((y-25/12)^2+(x-5)^2 = (25/12)^2, x = 2 .. 8, y = 0 .. 5, color = magenta);dr1 := draw([oo, Delta, varpi, N, M, J], printtext = true); dr2 := draw([alpha(color = black), lNJ(color = black), T1(color = green, filled = true), T2(color = green, filled = true)]);
inversion(M, M, C);
inversion(N, M, C);
Fig := proc (xOm)
local cir, c2, C2, C1, c3, C3, k, M, N, J, sol, dr, varpi;
global p, para, Para;
sol := solve(subs(x = xOm, para), y);
cir := (y-sol)^2+(x-xOm)^2 = sol^2; c2 := x^2+(y-3)^2 = 9;
geometry:-point(N, xOm, 0); sol := solve(subs(x = xOm, para), y);
geometry:-point(varpi, xOm, sol); k := 3/sol;
geometry:-point(M, xOm*k/(k+1), (3+k*sol)/(k+1));
geometry:-homothety(J, N, -k, M);
c3 := (x-(1/2)*xOm)^2+(y-3)^2 = (1/4)*dist(N, J)^2;
C1 := plots:-implicitplot(cir, x = -xOm .. 3*xOm, y = 0 .. 3*xOm, color = magenta);
C2 := plots:-implicitplot(c2, x = -xOm .. 2*xOm, y = 0 .. 2*xOm, color = blue);
C3 := plots:-implicitplot(c3, x = -xOm .. 2*xOm, y = 0 .. 2*xOm, color = blue);
dr := geometry:-draw([varpi, M, J]);
plots:-display([Para, C2, C1, C3, dr], view = [-xOm .. 3*xOm, -1 .. 3*xOm], axes = normal, scaling = constrained) end proc;

Fig(8);
display([seq(Fig(4+.8*i), i = 4 .. 15)]);
display({C1, Para, cir, dr1, dr2}, view = [-8 .. 8, -1 .. 8], axes = normal, scaling = constrained, size = [500, 500]);
I don't know what is that orthogonal circle to each tangent circles. Thank you to help me.

.

restart;
with(plots); with(LinearAlgebra);
_EnvHorizontalName := 'x';

_EnvVerticalName := 'y';

x1,y1,x2,y2,x3,y3:=0,-3,3,1,5,-2:   
A := [x1, y1]: B := [x2, y2]: C := [x3, y3]:

Barycentre := proc (A, B, t) description "Barycentre de 2 points A(1) et B(t) dans le rapport t";
return [(1-t)*A[1]+t*B[1], (1-t)*A[2]+t*B[2]] end proc;
ellip := proc (r1, r2) local a, b, c, d, e, f, D, E, F, eq1, eq2, eq3, eq4, eq5, eq6, x0, y0, EE, r3, sol, Ff, Tg;
global A, B, C;
r3 := -1/(r2*r1);
D := Barycentre(C, B, 1/(1-r1)); E := Barycentre(A, C, 1/(1-r2)); F := Barycentre(B, A, 1/(1-r3));
Ff := proc (x, y) options operator, arrow; a*x^2+2*b*x*y+c*y^2+2*d*x+2*e*y+f end proc;
Tg := proc (x0, y0, x, y) options operator, arrow; a*x*x0+b*(x*y0+y*x0)+c*y*y0+d*(x+x0)+e*(y+y0)+f end proc;
eq1 := Ff(D[1], D[2]);
eq2 := Ff(E[1], E[2]);
eq3 := Ff(F[1], F[2]);
eq4 := Tg(F[1], F[2], x1, y1);
eq5 := Tg(D[1], D[2], x2, y2);
eq6 := Tg(E[1], E[2], x3, y3);
sol := op(solve([eq1, eq2, eq3, eq4, eq5, eq6], [a, b, c, d, e]));
assign(sol);
EE := subs(f = 1, Ff(x, y) = 0) end proc;

ellip(-1, -7); tri := plot([A, B, C, A], color = blue):
 
po := plot([A, B, C], style = point, symbolsize = 15, symbol = solidcircle, color = red);
tp := textplot([[A[], "A"], [B[], "B"], [C[], "C"]], 'align' = {'above', 'left'});
x := 'x'; y := 'y';
ELL := seq(implicitplot(ellip(-7/11-(1/11)*j, -1/17-3*j*(1/17)), x = 0 .. 5, y = -3 .. 1, color = ColorTools:-Color([rand()/10^12, rand()/10^12, rand()/10^12])), j = 1 .. 17);
display([tri, ELL, po, tp], view = [-.5 .. 5.5, -4 .. 1.5], axes = none, scaling = constrained, size = [500, 500]);
Explore(implicitplot(ellip(r1, r2), x = 0 .. 5, y = -3 .. 1), parameters = [r1 = -2.18 .. -.7, r2 = -3 .. -.23]);
Can you tell me why this last instruction does't work ? Thank you.
 

We consider a triangle ABC, its circumscribed circle (O), of radius R, its inscribed circle (I) of centre I. We designate by α, β, γ the points of contact of BC, CA, AB with the circle (I), by A', B', C' the points of meeting other than A, B, C, of AI, BI, CI with the circle (O), by the media of BC, CA, AB.
.Establish that there is a conic (E), focus I, tangent to βγ, γα, αβ.
My code : 

restart;
with(geometry);
with(plots); _local(gamma);
_EnvHorizontalName := x; _EnvVerticalName := y;
alias(coor = coordinates);
point(A, -5, -5); point(B, 7, -1); point(C, 1, 5);
triangle(ABC, [A, B, C]); circumcircle(_O, ABC, 'centername' = OO); incircle(_I, ABC, 'centername' = Io);
line(lBC, [B, C]); sol := solve({Equation(_I), Equation(lBC)}, {x, y}); point(alpha, subs(sol, x), subs(sol, y));
line(lCA, [C, A]); sol := solve({Equation(_I), Equation(lCA)}, {x, y}); point(beta, subs(sol, x), subs(sol, y));
line(lAB, [A, B]); sol := solve({Equation(_I), Equation(lAB)}, {x, y}); point(gamma, subs(sol, x), subs(sol, y));
line(lAO, [A, OO]); intersection(Ap, lAO, lBC);
line(lBO, [B, OO]); intersection(Bp, lBO, lCA);
line(lCO, [C, OO]); intersection(Cp, lCO, lAB);
midpoint(l, B, C); midpoint(m, A, C); midpoint(n, A, B);
triangle(T, [alpha, beta, gamma]);
dr := draw([ABC(color = blue), _O(color = red), _I(color = magenta), lAO(color = black), lBO(color = black), lCO(color = black), T(color = red), alpha, beta, gamma, Ap, Bp, Cp, l, m, n], printtext = true);
display([dr], axes = normal, scaling = constrained, size = [800, 800]);
How to find the Equation of (E); Thank you.