## 400 Reputation

6 years, 338 days

## how to draw these 3 lines and then proje...

Maple 2018

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.

## How to show that these 2 procedures are ...

Maple 2018

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.

## How to find an orthogonal circle ? ...

Maple 2018

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.

.

## Why Explore does't work ?...

Maple 2018

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.

## Finding a conic tangents to 3 sites of a...

Maple 2018

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.

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