JAMET

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These are questions asked by JAMET

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.

I would like to show : in a quadrilateral circumscribed to an ellipse, the line passing through the middle of the diagonals passes through the centre of the ellipse.
My code is :

restart; with(geometry): with(plots): `local`(O):
_EnvHorizontalName := x: _EnvVerticalName := y:

alias(coor = coordinates):
ell := x^2/a^2+y^2/b^2 = 1:
point(P1,a*cos(omega), b*sin(omega)):
point(P2,a*cos(omega-(1/2)*Pi), b*sin(omega-(1/2)*Pi)): 
point(P3,a*cos(omega+(8/7)*Pi), b*sin(omega+(8/7)*Pi)):
point(P4,a*cos(omega+5*Pi*(1/2)), b*sin(omega+5*Pi*(1/2))):
a := 5: b := 3: omega := (1/5)*Pi:
Ell := implicitplot(ell, x = -a .. a, y = -b .. b, color = red): 
dr := draw([seq(P || k, k = 1 .. 4)], axes = normal, printtext = true):

for i from 1 to 4 do tgP||i := x*coor(P||i)[1]/a^2+y*coor(P||i)[2]/b^2 = 1 od:
poly := Matrix([coor(P1), coor(P2), coor(P3), coor(P4)]):
Quadri := polygonplot(poly, axes = normal, color = "DarkGreen", transparency = .8):

with(combinat): with(ListTools):
L := [1, 2, 3, 4]:
for i from 1 to 4 do Rotate(L, i)[1] od:
for i to 4 do solve({(tgP || Rotate)(L, i)[1], tgP || i}, {x, y}); point(S || i, subs(%, x), subs(%, y)); coor(S || i) end do;
Error, invalid input: subs received 1, which is not valid for its 1st argument
#otherwise
solve({tgP1, tgP2}, {x, y}): point(S1, subs(%, x), subs(%, y)); coor(S1):
                               S1
solve({tgP2, tgP3}, {x, y}): point(S2, subs(%, x), subs(%, y)); coor(S2):
                               S2
solve({tgP3, tgP4}, {x, y}): point(S3, subs(%, x), subs(%, y)); coor(S3):
                               S3
solve({tgP1, tgP4}, {x, y}): point(S4, subs(%, x), subs(%, y)); coor(S4):
                               S4

poly := Matrix([coor(S1), coor(S2), coor(S3), coor(S4)]):
Quadri2 := polygonplot(poly, axes = normal, color = "DarkGreen", transparency = .9):
#dr2:=draw(seq(S||k,k =1..4), axes = normal, printtext = true):
line(diag13, [S1, S3]): line(diag24, [S2, S4]): midpoint(M1, S1, S3): midpoint(M2, S4, S2): 
line(Lm, [M1, M2]):
dr2 := draw([S1, S2, S3, S4, M1, M2, Lm(color = black), diag13, diag24], axes = normal, printtext = true):
for i from 1 to 4 do
TgP||i := implicitplot(tgP||i, x = -a-5 .. a+5, y = -b-5 .. b+5, color = blue) od:
display([Ell, seq(TgP||i,i=1..4), Quadri, Quadri2,dr,dr2], view = [-a-5 .. a+3, -b-2 .. b+2], 
scaling = constrained, size = [700, 700]); Thank you for your answere.

How to find the location of the poles of a normal chord in an ellipse ?
Here is my code :

restart; with(geometry); with(plots); `local`(O);
_EnvHorizontalName := x; _EnvVerticalName := y;
corde := a*x/cos(theta)-b*y/sin(theta) = a^2-b^2;
isolate(corde, a/cos(theta));
Error, (in isolate) a*x/cos(theta)-b*y/sin(theta) = a^2-b^2 does not contain a/cos(theta)
eq1 := (a^2-b^2)*X/a^2 = a/cos(theta);
c := solve(eq1, cos(theta));
eq2 := (a^2-b^2)*Y/b^2 = -b/sin(theta);
s := solve(eq2, sin(theta));
lieu := simplify(expand((a^2-b^2)^2*X^2*Y^2*(c^2+s^2 = 1)));
allvalues(eliminate({eq1, eq2}, theta))[1][2];
ell := x^2/a^2+y^2/b^2 = 1;
P := [a*cos(theta), b*sin(theta)];
tgP := x*P[1]/a^2+y*P[2]/b^2 = 1;

sol := solve({corde, ell}, {x, y});
tgP1 := simplify(x*rhs(sol[2][1])/a^2+y*rhs(sol[2][2])/b^2 = 1);

Drawing in a case
a := 5; b := 3; theta := (1/6)*Pi;

line(l1, corde); conic(co, ell);
Pole(P1, l1, co); coordinates(P1);
a := 5; b := 3; theta := (1/6)*Pi;
Ell := implicitplot(ell, x = -a .. a, y = -b .. b, color = red);
Cor := implicitplot(corde, x = -a-1 .. a, y = -b-1 .. b, color = blue);
TgP := implicitplot(tgP, x = 0 .. 10, y = -5 .. 10, color = magenta);
TgP1 := implicitplot(tgP1, x = -5 .. 10, y = -5 .. 10, color = magenta);
lieu := subs(X = x, Y = y, lieu);
subs(x = 125*sqrt(3)*(1/24), y = -27/8, lieu);
Lieu := implicitplot(lieu, x = -a .. a, y = -b .. b, color = green);
dr := draw(P1);
display([Ell, Cor, Lieu, TgP, TgP1, dr], axes = normal, view = [-10 .. 10, -10 .. 10], scaling = constrained);

Why the drawin of the location (lieu) does not appear ? Thank you.

I am solving this question :the line joining the ends of 2 rectangular diameters of an ellipse, remains tangent to a fixed circumference. My code is :

restart; with(geometry); with(plots); unprotect(O);
_EnvHorizontalName := x; _EnvVerticalName := y;
ell := x^2/a^2+y^2/b^2 = 1;
a := 5; b := 3; alpha := (1/6)*Pi; p := sqrt(a^2*b^2/(a^2+b^2));
PQ := x*cos(alpha)+y*sin(alpha)-p; drPQ := solve(PQ, y);
OPQ := x^2/a^2+y^2/b^2-((x*cos(alpha)+y*sin(alpha))/p)^2;
sol := solve({OPQ, ell}, {x, y}, explicit); P := [subs(sol[1], x), subs(sol[1], y)]; Q := [subs(sol[3], x), subs(sol[3], y)];
O := [0, 0];
Ell := implicitplot(ell, x = -a .. a, y = -b .. b, color = red);
DrOPQ := implicitplot(OPQ, x = -a .. a, y = -b .. b, color = magenta, numpoints = 5000);
DrPQ := plot(drPQ, x = -6 .. 6, color = green);
line(OP, 2*x-y); line(OQ, -(1/2)*x-y);

Points := pointplot([O[],P[],Q[]], symbol = solidcircle, color = red, symbolsize = 10):

T := textplot([[O[], "O"],[P[],"P"],[Q[],"Q"]], font = [times, 15], align = {below, right}):
cir := x^2+y^2 = p^2;
Cir := implicitplot(cir, x = -a .. a, y = -b .. b, color = black);
display([Ell, Cir, DrPQ, DrOPQ, Points, T], view = [-6 .. 6, -4 .. 6], axes = normal, scaling = constrained);
Fig := proc (k) local alpha, PQ, drPQ, DrPQ, OPQ, DrOPQ, sol, P, Q, Points, T; global a, b, p, ell, Ell, Cir; alpha := k; PQ := x*cos(alpha)+y*sin(alpha)+p; drPQ := solve(PQ, y); OPQ := x^2/a^2+y^2/b^2-(x*cos(alpha)+y*sin(alpha))^2/p^2; sol := solve({ell, OPQ}, {x, y}, explicit); P := [subs(sol[1], x), subs(sol[1], y)]; Q := [subs(sol[3], x), subs(sol[3], y)]; Points := pointplot([P[], Q[]], symbol = solidcircle, color = red, symbolsize = 10);
T := textplot([[P[], "P"], [Q[], "Q"]], font = [times, 15], align = {below, right}); DrPQ := plot(drPQ, x = -6 .. 6, color = green);µ DrOPQ := implicitplot(OPQ, x = -a .. a, y = -b .. b, color = magenta, numpoints = 5000);
display([Ell, Cir, DrPQ, DrOPQ, Points, T], view = [-a .. a, -b .. b], axes = normal, scaling = constrained) end proc;

Fig((1/4)*Pi);
Error, (in Fig) invalid subscript selector
nframes := 100; plots:-display([seq(Fig(2*Pi*i/nframes), i = 0 .. nframes)], insequence, scaling = constrained);
Error, (in Fig) invalid subscript selector
Explore(Fig(n), n = 0 .. 2*Pi);
Thank you for your help.

We give a line (D) and a point A located at a distance AH=h from D. A constant angle of magnitude alpha pivots to its apex A and we call B and C the points where its sides cut the line D. Let O be the center of the circle circumscribed to the triangle ABC.
Demonstrate that the B and C tangents to the O circle keep a fixed direction. 
Here is my code which don't work for slopes are not equal.

restart; with(plots): with(geometry):unprotect(D):
_EnvHorizontalName := 'x':_EnvVerticalName := 'y':
line(D, y = (1/2)*x-1); point(A, 5, 5); PerpendicularLine(lp, A, D); h := distance(A, D); intersection(H, D, lp);
alpha := (1/16)*Pi;
rotation(lp1, lp, (1/6)*Pi, 'clockwise', A); rotation(lp2, lp1, (1/6)*Pi-alpha, 'clockwise', A); FindAngle(lp1, lp2); evalf(%);
intersection(B, D, lp1); intersection(C, D, lp2);
triangle(T, [A, B, C]);
circumcircle(Elc, T, 'centername' = OO);
TangentLine(tgB, B, Elc); TangentLine(tgC, C, Elc);
evalf(slope(tgB)); evalf(slope(tgC));
dr := draw([D(color = blue), lp(color = red), Elc(color = green), A, B, C, T(color = black), H, tgB, tgC], printtext = true);

display([dr], axes = none, scaling = constrained);
Fig := proc (k) local dr, Elc, B, C, lp1, lp2; global D, A, lp, H, alpha; geometry:-rotation(lp1, lp, (1/6)*Pi+k, 'clockwise', A); geometry:-rotation(lp2, lp1, (1/6)*Pi-alpha+k, 'clockwise', A); geometry:-intersection(B, D, lp1); geometry:-intersection(C, D, lp2); geometry:-triangle(T, [A, B, C]); geometry:-circumcircle(Elc, T, 'centername' = OO); geometry:-TangentLine(tgB, B, Elc); geometry:-TangentLine(tgC, C, Elc); dr := geometry:-draw([D(color = blue), lp(color = red), Elc(color = green), A, B, C, T(color = black), H, tgB, tgC], printtext = true); plots:-display([dr], axes = none, scaling = constrained) end proc;
iframes := 10;

display([seq(Fig((1/12)*Pi+i/(10*iframes)), i = 1 .. iframes)], insequence, scaling = constrained);
How to improve this code ? Thank you.

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