7 years, 0 days

## Monge'scircle...

@tomleslie  Very good prg using geomety, Thang you very much;

## Barycenter...

@sand15
Yes, weights may be negative

## Tomlesie, I thank you yout program is ve...

Tomlesie, I thank you yout program is very good.
I want now to draw equilateral hyperbole
orthocenter(H, Tr1):
evalf(coordinates(H));
#conic(Hyp,[P1,P2,P3,P4,H],[x,y]);
But the computer crashes
Perhaps we should model the equation of this hyperbole?

## How to place points OO, G , E (Euler cen...

Very good prg
line(L1,[P1,P3]):
line(L2,[P1,P4]):
ntersection(G,L1,L2);#gravity center of P1P2P3P4
G1:=convert(coordinates(G),Vector);
G:=<<-(5*sqrt(2))/2>|<(5*sqrt(2))/2>>;
pointplot(G,color = blue, symbol = diamond)
E - Euler center of P1,P2,P3,P4
I don(t know how yo place these points.

## difficulty with polygonplot...

#p1 := polygonplot([P1,P2,P3,P4], color = blue, filled=true,transparency = 0.7):
#p2 := polygonplot([o1,o2,o3,o4], color = red, filled=true,transparency = 0.8):
display
( [ draw
( [ seq
( [ `||`(P, j)(color = black, symbol = solidcircle, symbolsize = 12),
`||`(o, j)(color = black, symbol = solidcircle, symbolsize = 12),
`||`(Tr, j)(color = green),
`||`(Elc, j),
`||`(seg, j)(color=black,filled=true)

][],
j=1..4
),
cir(color = blue)#p1,p2
When i take off #, I got massage of error with polygonplot. How to correct. Thank you.

## I am sorry. I got still these messages &...

I am sorry. I got still these messages

triangle: 0 < -P1+P2+P4
Error, (in geometry:-triangle) not enough information: the triangle is not defined
Error, (in geometry:-EulerCircle) wrong type of argument, expecting a triangle
triangle: 0 < -P1+P2+P3
Error, (in geometry:-triangle) not enough information: the triangle is not defined
Error, (in geometry:-EulerCircle) wrong type of argument, expecting a triangle
triangle: 0 < -P3+P2+P4
Error, (in geometry:-triangle) not enough information: the triangle is not defined
Error, (in geometry:-EulerCircle) wrong type of argument, expecting a triangle
triangle: 0 < -P1+P3+P4
Error, (in geometry:-triangle) not enough information: the triangle is not defined
Error, (in geometry:-EulerCircle) wrong type of argument, expecting a triangle
cir

ngon1 := [P1, P2, P3, P4]

Error, (in geometry:-coordinates) wrong type of argument
ngon2 := [P1, P2, P3, P4]

Error, (in geometry:-coordinates) wrong type of argument
Error, (in geometry:-coordinates) wrong type of argument

Sorry I need still your help. Could we still better use the computer to condense the reports ( 1,2,3,4) ?

restart:
with(geometry):
with(plots):
_EnvHorizontalName = 'x':
_EnvVerticalName = 'y':
R := 5:
ang := [3/4*Pi, -(3*Pi)/4, -Pi/6,4*Pi/9]:
seq
( point
( `||`(P, i),
[ R*cos(ang[i]), R*sin(ang[i])]
),
i = 1 .. 4
):
seq
( dsegment
( `||`(seg, i),
[ `||`(P, i),
`||`(P, irem(i, 4) + 1)
]
),
i = 1 .. 4
):
triangle(Tr1,[P1,P2,P4]):
EulerCircle(Elc1,Tr1,'centername'=o1):
triangle(Tr2,[P1,P2,P3]):
EulerCircle(Elc2,Tr2,'centername'=o2):
triangle(Tr3,[P3,P2,P4]):
EulerCircle(Elc3,Tr3,'centername'=o3):
triangle(Tr4,[P1,P3,P4]):
EulerCircle(Elc4,Tr4,'centername'=o4):
circle(cir, [point(OO, [0, 0]), R]):
ngon1 :=[P1,P2,P3,P4]:
poly1:=polygonplot(ngon1,color=blue,filled=true):
ngon2 :=[P1,P2,P3,P4]:
poly2:=polygonplot(ngon2,color=yellow,filled=true):
display
( [ draw
( [ P1(color = black, symbol = solidcircle, symbolsize = 12),
P2(color = black, symbol = solidcircle, symbolsize = 12),
P3(color = black, symbol = solidcircle, symbolsize = 12),
P4(color = black, symbol = solidcircle, symbolsize = 12),
o1(color = black, symbol = solidcircle, symbolsize = 12),
o2(color = black, symbol = solidcircle, symbolsize = 12),
o3(color = black, symbol = solidcircle, symbolsize = 12),
o4(color = black, symbol = solidcircle, symbolsize = 12),
seg1,
seg2,
seg3,
seg4,
Tr1(color=green),Tr2(color=green),Tr3(color=green),Tr4(color=green),
Elc1,Elc2,Elc3,Elc4,
cir(color = blue)
]
),
textplot
( [ seq
( [ coordinates(`||`(P, i))[],
convert(`||`(P, i), string)
],
i=1..4
)
],
[ seq
( [ coordinates(`||`(o, i))[],
convert(`||`(o, i), string)
],
i=1..4
)
],

align=[above, right]
)
],
axes=none
);

Error, (in plots:-polygonplot) incorrect arguments for creating polygons structure
Error, (in plots:-polygonplot) incorrect arguments for creating polygons structure
Error, (in plots:-textplot) unexpected option: [[(1/2)*((125/4)*2^(1/2)+((5/4)*2^(1/2)+(5/2)*sin((4/9)*Pi))^2*(-(5/4)*2^(1/2)+(5/2)*sin((4/9)*Pi))-((5/4)*2^(1/2)+(5/2)*sin((4/9)*Pi))*(-(5/4)*2^(1/2)+(5/2)*cos((4/9)*Pi))^2-((5/4)*2^(1/2)+(5/2)*sin((4/9)*Pi))*(-(5/4)*2^(1/2)+(5/2)*sin((4/9)*Pi))^2+(-(5/4)*2^(1/2)+(5/2)*cos((4/9)*Pi))^2*(-(5/4)*2^(1/2)+(5/2)*sin((4/9)*Pi)))/(-25/4-(25/4)*2^(1/2)*cos((4/9)*Pi)), -(1/2)*((5/2)*2^(1/2)*((-(5/4)*2^(1/2)+(5/2)*cos((4/9)*Pi))^2+((5/4)*2^(1/2)+(5/2)*sin((4/9)*Pi))^2)-(5/2)*2^(1/2)*((-(5/4)*2^(1/2)+(5/2)*cos((4/9)*Pi))^2+(-(5/4)*2^(1/2)+(5/2)*sin((4/9)*Pi))^2)-(-(5/4)*2^(1/2) ... 2^(1/2)*sin((4/9)*Pi)+(25/16)*2^(1/2)*3^(1/2)-(25/16)*2^(1/2)), "o4"]]

Thank you very much for your help.

## how to solve the system ?...

solve({x = 1 + 5*t, y = 1 + 3*t, 5*x + 3*y + 1 = 0}, {t, x, y});
Warning, solving for expressions other than names or functions is not recommended.
Thank you.

## Euler and Poncelet's Triangles...

Thank you very much. How to leave fixed the circumscribed triangle P1P2P2 and make mobile the triangle inscribed AM2 M3 ? Then add the letters on figure M1, M2, A. Note that the M1M2 segment remains tangent to the inscribed circle.

## Projection of a point on a line...

ProjPL:=proc(P,A,B)
local H,x1,x2,x33,y1,y2,y3,t:
A:= <x1,y1>;
B:= <x2,y2>;
P :=<x3, y3>;
H:=A+t*(B-A); solve( (P-H)^+ . (B-A), t );
'H':=eval(H, t=`%` );
end:
Error, (in ProjPL) invalid left hand side of assignment/ How to correct this error ? . Thank you.

## Location of T...

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

## I am trying to animate the figure. it s...

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

## property of an ellipse...

#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.

## problem with P remaining on the upper se...

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.

## difficulty with parallelline...

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.

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