Kitonum

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11 years, 355 days

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These are replies submitted by Kitonum

JAMET wrote:
"Let Q be a quadrilateral which has a inscribed circle and a circumscribed circle. Show that the centers of these 2 circles and the point of intersection of the diagonals of the quadrilateral are aligned."   

The proof of this statement for the general case can be read here  http://mathhelpplanet.com/viewtopic.php?f=28&t=68666   ( chebo's post, in Russian).

@Vrighty  It seems you did not understand me. When you are not using indexes, you should use eval command with your approach. With indexes you can not use eval:

restart;
F(y):=A+B+C;
A:=10*y;
B:=y²-4;
C:=5-y;
F(y):=eval(F(y));

F[y]:=A+B+C;
A:=10*y;
B:=y²-4;
C:=5-y;
F[y]:=F[y]; 

                             

 

@Thomas Dean  I do not know how to do this programmatically in the general case. You see that, as Thomas Richard pointed out, Maple 2020 successfully handled this example. Maybe one of the Maple developers will answer your questions.

@Carl Love  Thank you for your detailed analysis (vote up). 

Since this set of points on the plane is described by the polynomial equation  P(x)=0 ,
where  P(x)=x^8+4*x^6*y^2+6*x^4*y^4+4*x^2*y^6+y^8-12*x^7-36*x^5*y^2-36*x^3*y^4-12*x*y^6+252*x^6+324*x^4*y^2-108*x^2*y^4-180*y^6-5184*x^3*y^2-5184*x*y^4+7776*x^4+23328*y^4+116640*x^3+209952*x*y^2+839808*x^2+2519424*x+3779136-1944^2 ,
it would be great if we could split this polynomial  P(x)  into the product of 3 polynomials corresponding to each connected curve.

 

@Carl Love  It seems to me a manifestation of some disrespect for the participants of this forum who do not have Maple 2019 (I think this applies to most of them), the writing a code that obviously does not work in Maple <2019. 

Here is almost the same code, but it also works for Maple <2019. Since Fibonacci numbers are growing very fast, the code works great for extremely large ranges (I took 10^10000 instead of 10000):

restart;
a,b:= 0,1:  
while b<10^10000 do
s:=a+b; a,b:= b,s;  
if issqr(b) then print(b) fi;
od:

                                     1
                                    144

  

Where is the system itself? Copy it here in text form (not a picture).

@goli  The  explicit  option allows you to get the roots of an equation in explicit form, of course, if possible. See acer's answer  for  TT8 .

@goli  In the first example

RootOf(_Z^2*l^2+3*_Z^4-3)

, we have a simple biquadratic equation, the roots of which are easily expressed in terms of the coefficients of this equation. The equation in your example (I named it  A ) is much more complicated and probably just there is no formula expressing the roots of this equation in terms of its coefficients. As you can see, allvalues command does not help. Only when specifying the parameter value  do we get the solution explicitly.


 

restart;
A:=RootOf(6*_Z^3+(27+3*RootOf(_Z^2*l^2+3*_Z^4-3)^2*sqrt(9-3*RootOf(_Z^2*l^2+3*_Z^4-3)^2*l^2))*_Z^2+(3*sqrt(9-3*RootOf(_Z^2*l^2+3*_Z^4-3)^2*l^2)*l^4*RootOf(_Z^2*l^2+3*_Z^4-3)^2-9*sqrt(9-3*RootOf(_Z^2*l^2+3*_Z^4-3)^2*l^2)*l^2+90*RootOf(_Z^2*l^2+3*_Z^4-3)^2*l^2-18*l^4+6*l^6*RootOf(_Z^2*l^2+3*_Z^4-3)^2-81+45*RootOf(_Z^2*l^2+3*_Z^4-3)^2*sqrt(9-3*RootOf(_Z^2*l^2+3*_Z^4-3)^2*l^2))*_Z-324-3*l^8+l^10*RootOf(_Z^2*l^2+3*_Z^4-3)^2+108*RootOf(_Z^2*l^2+3*_Z^4-3)^2*sqrt(9-3*RootOf(_Z^2*l^2+3*_Z^4-3)^2*l^2)-3*sqrt(9-3*RootOf(_Z^2*l^2+3*_Z^4-3)^2*l^2)*l^6+sqrt(9-3*RootOf(_Z^2*l^2+3*_Z^4-3)^2*l^2)*l^8*RootOf(_Z^2*l^2+3*_Z^4-3)^2-63*sqrt(9-3*RootOf(_Z^2*l^2+3*_Z^4-3)^2*l^2)*l^2+30*sqrt(9-3*RootOf(_Z^2*l^2+3*_Z^4-3)^2*l^2)*l^4*RootOf(_Z^2*l^2+3*_Z^4-3)^2+45*l^6*RootOf(_Z^2*l^2+3*_Z^4-3)^2+351*RootOf(_Z^2*l^2+3*_Z^4-3)^2*l^2-108*l^4, index = 1):
Sol:=allvalues(A);

RootOf(72*_Z^3+(-3*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^2+3*(l^4+36)^(1/2)*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)+324)*_Z^2+(-12*l^8+12*(l^4+36)^(1/2)*l^6-3*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^6+3*(l^4+36)^(1/2)*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^4-396*l^4+180*(l^4+36)^(1/2)*l^2-99*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^2+45*(l^4+36)^(1/2)*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)-972)*_Z-2*l^12+2*(l^4+36)^(1/2)*l^10-(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^10+(l^4+36)^(1/2)*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^8-126*l^8+90*(l^4+36)^(1/2)*l^6-48*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^6+30*(l^4+36)^(1/2)*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^4-1998*l^4+702*(l^4+36)^(1/2)*l^2-486*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^2+108*(l^4+36)^(1/2)*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)-3888, index = 1), RootOf(72*_Z^3+(-3*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^2+3*(l^4+36)^(1/2)*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)+324)*_Z^2+(-12*l^8+12*(l^4+36)^(1/2)*l^6-3*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^6+3*(l^4+36)^(1/2)*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^4-396*l^4+180*(l^4+36)^(1/2)*l^2-99*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^2+45*(l^4+36)^(1/2)*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)-972)*_Z-2*l^12+2*(l^4+36)^(1/2)*l^10-(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^10+(l^4+36)^(1/2)*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^8-126*l^8+90*(l^4+36)^(1/2)*l^6-48*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^6+30*(l^4+36)^(1/2)*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^4-1998*l^4+702*(l^4+36)^(1/2)*l^2-486*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^2+108*(l^4+36)^(1/2)*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)-3888, index = 1), RootOf(72*_Z^3+(-3*(36+2*l^4+2*(l^4+36)^(1/2)*l^2)^(1/2)*l^2-3*(l^4+36)^(1/2)*(36+2*l^4+2*(l^4+36)^(1/2)*l^2)^(1/2)+324)*_Z^2+(-12*l^8-12*(l^4+36)^(1/2)*l^6-3*(36+2*l^4+2*(l^4+36)^(1/2)*l^2)^(1/2)*l^6-3*(l^4+36)^(1/2)*(36+2*l^4+2*(l^4+36)^(1/2)*l^2)^(1/2)*l^4-396*l^4-180*(l^4+36)^(1/2)*l^2-99*(36+2*l^4+2*(l^4+36)^(1/2)*l^2)^(1/2)*l^2-45*(l^4+36)^(1/2)*(36+2*l^4+2*(l^4+36)^(1/2)*l^2)^(1/2)-972)*_Z-2*l^12-2*(l^4+36)^(1/2)*l^10-(36+2*l^4+2*(l^4+36)^(1/2)*l^2)^(1/2)*l^10-(l^4+36)^(1/2)*(36+2*l^4+2*(l^4+36)^(1/2)*l^2)^(1/2)*l^8-126*l^8-90*(l^4+36)^(1/2)*l^6-48*(36+2*l^4+2*(l^4+36)^(1/2)*l^2)^(1/2)*l^6-30*(l^4+36)^(1/2)*(36+2*l^4+2*(l^4+36)^(1/2)*l^2)^(1/2)*l^4-1998*l^4-702*(l^4+36)^(1/2)*l^2-486*(36+2*l^4+2*(l^4+36)^(1/2)*l^2)^(1/2)*l^2-108*(l^4+36)^(1/2)*(36+2*l^4+2*(l^4+36)^(1/2)*l^2)^(1/2)-3888, index = 1), RootOf(72*_Z^3+(-3*(36+2*l^4+2*(l^4+36)^(1/2)*l^2)^(1/2)*l^2-3*(l^4+36)^(1/2)*(36+2*l^4+2*(l^4+36)^(1/2)*l^2)^(1/2)+324)*_Z^2+(-12*l^8-12*(l^4+36)^(1/2)*l^6-3*(36+2*l^4+2*(l^4+36)^(1/2)*l^2)^(1/2)*l^6-3*(l^4+36)^(1/2)*(36+2*l^4+2*(l^4+36)^(1/2)*l^2)^(1/2)*l^4-396*l^4-180*(l^4+36)^(1/2)*l^2-99*(36+2*l^4+2*(l^4+36)^(1/2)*l^2)^(1/2)*l^2-45*(l^4+36)^(1/2)*(36+2*l^4+2*(l^4+36)^(1/2)*l^2)^(1/2)-972)*_Z-2*l^12-2*(l^4+36)^(1/2)*l^10-(36+2*l^4+2*(l^4+36)^(1/2)*l^2)^(1/2)*l^10-(l^4+36)^(1/2)*(36+2*l^4+2*(l^4+36)^(1/2)*l^2)^(1/2)*l^8-126*l^8-90*(l^4+36)^(1/2)*l^6-48*(36+2*l^4+2*(l^4+36)^(1/2)*l^2)^(1/2)*l^6-30*(l^4+36)^(1/2)*(36+2*l^4+2*(l^4+36)^(1/2)*l^2)^(1/2)*l^4-1998*l^4-702*(l^4+36)^(1/2)*l^2-486*(36+2*l^4+2*(l^4+36)^(1/2)*l^2)^(1/2)*l^2-108*(l^4+36)^(1/2)*(36+2*l^4+2*(l^4+36)^(1/2)*l^2)^(1/2)-3888, index = 1)

(1)

nops([%]);
allvalues(Sol[1]);

4

 

RootOf(72*_Z^3+(-3*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^2+3*(l^4+36)^(1/2)*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)+324)*_Z^2+(-12*l^8+12*(l^4+36)^(1/2)*l^6-3*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^6+3*(l^4+36)^(1/2)*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^4-396*l^4+180*(l^4+36)^(1/2)*l^2-99*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^2+45*(l^4+36)^(1/2)*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)-972)*_Z-2*l^12+2*(l^4+36)^(1/2)*l^10-(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^10+(l^4+36)^(1/2)*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^8-126*l^8+90*(l^4+36)^(1/2)*l^6-48*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^6+30*(l^4+36)^(1/2)*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^4-1998*l^4+702*(l^4+36)^(1/2)*l^2-486*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^2+108*(l^4+36)^(1/2)*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)-3888, index = 1)

(2)

 


 

Download roots.mw

@tomleslie  Perfect solution. I also tried to use Fractals  package, but did not understand anything in the help. In my opinion, it is written too concisely and is very difficult to understand.
Just one note about your code - the animation is too fast. The minimum change in the last line of code allows you to make it 10 times slower:

display( [seq( doKoch(j)$10, j=0..5)], insequence, size=[500,500]);

                    

 

@Carl Love  I get it. But apparently your method is only suitable for Triangles in VectorCalculus:-int :

restart;

J:=VectorCalculus:-int(1, [x, y] = Rectangle(0 .. (1/2)*Pi, 0 .. (1/2)*Pi), inert):
value(J);

plots:-display(
    [seq](
        plot3d(0, op([1,2],j), op(2,j)),
        j= indets(J, Int(Int(algebraic, name= range(algebraic)), name= range(numeric)))
    ),
    orientation= [180, 0, 180]
);

 

@Carl Love  For triangles in  VectorCalculus:-int , your new code works. But when I tried to apply it to the same original integral, written in the usual way, an error again appears:

restart;

J:=Int(x*y,[y=0..1-x,x=0..1]):
value(J);

plots:-display(
    [seq](
        plot3d(0, op([1,2],j), op(2,j)),
        j= indets(J, Int(Int(algebraic, name= range(algebraic)), name= range(numeric)))
    ),
    orientation= [180, 0, 180]
);

 

@Carl Love  Unfortunately, this does not always work:

restart;
J:= VectorCalculus:-int(x*y, [x,y]= Triangle(<1/2,0>, <1,0>, <0,1>), inert):
plot3d(0, op([1,2], J), op(2,J), orientation= [180,0,180]);

  Error, (in plot3d) bad range arguments: x = 0 .. 1/2, -10. .. 10.
 

You should upload your worksheet here using the bold green up-arrow in the mapleprimes editor, or at least paste your complete code in text form (not a picture).

@mmcdara  You wrote "...you are not working with samples." But we can quite consider these specific data  X  and  Y  as a kind of sample. Of course, your interpretation is more natural and accurate, since when calculating the integral of an explicitly given function, information about the behavior of the function at an infinite number of points (over the entire segment) is used. However, the final results are very close.

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