@Preben Alsholm Thank you.  There is still something I don't quiet understand yet. I see if I didn't specifically use P3 in 1st proc , I get the inorrect output.  I know that would mean redundant input whic is not good.

foo2:=overload([                   
                  proc(P1::list,P2::list,P3::list,$)
                      option overload(callseq_only);
                   return  [P1,P2,P3],"3lists";
                  end proc,
                  proc(P1::list,P2::list,a::algebraic:=4,$)
                      option overload;
                     
                   return  [P1,P2],"2 lists"
                  end proc
                       ]):
foo2([1,2],[3,4]); 
foo2([1,2],[3,4],5);
foo2([1,2],[3,4],[4,7]);
                  [[1, 2], [3, 4]], "2 lists"

                  [[1, 2], [3, 4]], "2 lists"

               [[1, 2], [3, 4], [4, 7]], "3lists"

foo3:=overload([                   
                  proc(P1::list,P2::list,P3::list,$)
                      option overload(callseq_only);
                     #2*P3; # using this line it all work correctly;
                   return  "3lists";
                  end proc,
                  proc(P1::list,P2::list,a::algebraic:=4,$)
                      option overload;
                     
                   return  "2 lists"
                  end proc
                       ]):
foo3([1,2],[3,4]); #incorrect output;
foo3([1,2],[3,4],5);
foo3([1,2],[3,4],[4,7]);
                            "3lists"

@JAMET It would be much easier for people to help you if you upload your worksheet/document., 

Why all theinverted commas "............."?

There are numerious  places semicolins are missing.

In you plotting for F1 remove what is highlighted in red

 F1(color = black, symbol = soli4*yM2, yM1^2*dcircle, symbolsize=12)

then it works

@rlopez Isee you also posted something on Marden's Theoren.

@Carl Love This best example I have righrt now is in my reply here to a question How to draw an ellipse cut by lies - MaplePrimes  This made extensive use of factrix.   

Download 2023-12-23_A_Pascal_Conic_parameterised.mw

@Carl Love General Euclidean domain with rational functions eg parameterisation of a circle 

[(1-t^2)/(1+t^2) , 2t/(1+t^2 , 1]

Also there are square roots. I am playing around a lot with conic sections so plenty of square roots popup.

@Carl Love I will test this when at home over the next day or two. I was wondendering about approaching the problem using gcd and fold. Did't try anything yet. Can 'igcd' and 'ilcm' be replaced with 'gcd' and 'lcm'. The reason I ask is I mostly use factrix for projective geometry. Often  symbolic expressions are in the vectors, where row form represents a point and column form a line. Rescaling is really useful in that application.

  Didn't know I couldn't repost it here. Thanks for the link to the Maple Advisor Database.

@Carl Love Thank you. I confess I do not understand the code.e.g. Why do you start off with __M ?

 @C_R @mmcdara ​​​​​​​@acer ​​​​​​​  All good answers, I was playing around with my answer to this quastion on Pascals Theorem How to draw an ellipse cut by lies - MaplePrimes    to represent the conics as matrices. Sometimes its nicer to have e.g. x^2+y^2=R^2 and at times x^2+y^2-R^2 is just better. One of the procedures can now handle either automatically.  Good the see all the different apraoches. Mixed collections my well turn up too.

@rlopez Well I had always wondered about "unapply" and why it was called that.

How does one actually find out about these new names  / enchancements?  Are there others like this in Maple 2023?

I find the topic interesting but have not had time to explore it. I see you have had several tries at this. This is a link to a PhD thesis. Hope it helps you.

Creation of Hyperbolic Ornaments (tum.de)

Edit:- I traced the above down from this question on Math Stackexchange

hyperbolic geometry - Tiling on Poincaré disc - Mathematics Stack Exchange

@Thomas Richard  Yes that is it.

@mmcdara This is a nice concept. Shall use it at some point. Meanwhile alll is working now nicely. 

@acer Thank you. Have usually in the past used "insequence=true". I didn't know one could just use "insequence".

I see what you mean on "animate" using non integer values of i.  Not an issue in this case.

I asked a similar type question last year. I dont remember all the details of how I got the problem solved.

There might be something useful in the replies

Select powers and coeffs on a general type polynomial. - MaplePrimes