@tomleslie 
G is the centroid of the (inscribed) quadrilateral P1P2P3P4.
E is the Euler center of  the quadrilateral P1P2P3P4  defined as the point in the intersection of the four Euler circles of the triangles P1P2P3, ..., P4P1P2. (E exists!).

Note that OG = (OP1+OP2+OP3+OP4)/4, OE = (OP1+OP2+OP3+OP4)/2,  OPk  being vectors.

@Alex0099  1. The order of summation does not matter, the summand being symmetric.

2. The proof could be very tricky. Probably here the Residue Theorem should be useful.  

@Thomas Dean  In Windows, after a restart everything is OK. Sorry, I don't use Linux.

@emendes  procedure is a type, so, its use is covered.

AFAIK the Draghilev method is to find a natural parametric representation of a curve given as an intersection of n surfaces in (n+1) unknowns. That is what Rouben did, but using direct computations.

(I was never able to find the original article of Draghilev, but there are many places -- even in this site -- where the method is used).   

@mmcdara An old question of mine:

Order in plots[display] - MaplePrimes

@lcz Replace:

first:= fsolve((f(t)-x0)^2+(g(t)-y0)^2=0.02^2, t=0..1);
last := fsolve((f(t)-x3)^2+(g(t)-y3)^2=0.02^2, t=0..1);

@Stretto I don't see how you could make ArgK to work. As it is, after correcting the Is problem (see acer's comment), it would work only when computing the values in an array, but not randomly (in any language!). I'd suggest to take the dsolve approach.

@Carl Love  A and B having polynomial entries, A.B = Id ==> A invertible, that's all.
BTW, your potential objection is also valit for the provided answer, if A = <p,0;0,1/p>, where det(A) gives 1. But if we disect such a simple problem, everything becomes complicated (more or less artificially).

Actually, you solved the problem! For a)  it's enough to compute  simplify(A . B) , obtaining the unit matrix, without any assumption on p. (Of course it's very useful to know what a singular matrix is.)

@fhn2022 A smooth discontinuous function? Analytic? In mathematics such things do not exist
[f analytic ==> f smooth ==> f continuous].

@fhn2022 Why are you so sure that the function is continuous? You have many complex square roots! Denominators too!

@fhn2022 You may need to increase Digits but then even a longer time will be needed.
If the problem is important for you, try to optimize/simplify the computations.

I just want to mention that a similar problem is here:

https://www.mapleprimes.com/questions/222071-Drawing-An-Apollonian-Gasket#answer239140