@Doug Meade 

You have assumed in the worksheet that x,y do not depend on a.
It seems to me that this is impossible.
So, actually x = x(a), y=y(a).

A numerical investigation will be very difficult.
Note that x(a)=y(a)=1/2, F(a)=1 almost satisfy the system, because
[eq1,eq2,eq3] reduces to

[ 10^(-53)*(1+a)^3 = 0, 0=0, 0=0].

@Kitonum

An irritating fact is that

a:=0;
plots[inequal](
sqrt((x-1)^2+(y-1)^2 ) < sqrt ((x+1)^2+(y+1)^2 +a),
x=-5..5, y=-5..5, color="LightGreen");

does not work properly, but it is ok for any a<>0. Maple can be so unpredictable sometimes!

Of course kitonum's solution does not contain proofs. My point was that each step can be easily formulated with full mathematical rigor. I am sorry if you don't see this, it's not my fault.

@Markiyan Hirnyk 

kitonum's solution is rigorous. The regions are actually the connected components of the plane \ {five lines}.
In a connected set, the signum of a continuos <>0 function is constant. Please don't try to convince me about an elementary fact in mathematical analysis because I work and teach in this domain for a long time.

@Markiyan Hirnyk 

Yes, I know, I am never right :-).

But please note that kitonum's solution (which is btw very nice) is practically a "by hand" one.
The regions can be drawn by hand at once, and the inequalities in the chosen points can also be verified easily becasuse the initial inequality is equivalent to

in those points.

This type of problems are much easier to solve by hand than using Maple.
It would be possible to give a Maple solution, but only imitating the "by hand" one (which must be done by the user).

@asa12 

You should explain mathematically what wou want to obtain. Without Maple code!

The Groebner bases are related to the ideal generated by a set of polynomials. If you remove polynomials from a basis, the ideal is changed. What theoretical facts are you intending to use?

simplify needs occasionally to use evalf (for localizing rootofs etc).
If Digits is raised to 15 (for ee, or even to 11 for u), the bug disappears!

@tejolson 

evalf(u) seems to be ok here. It's only roundoff error ("catastrophic cancellation"):

evalf([op(numer(u))]);

But the bug is from acer's version. I don't see any problem with your a,b,c, except that they are a bit difficult to simplify.

@acer 

Could you tell us the source of this example and where is the bug?
(I recall that in some old Maple texts, simplify without any option was not considered very safe.)

Edit. A simplified version:

u:=(4*(28412308*I*sqrt(3)+44068556*sqrt(3)-76328978-49211561*I))/((-3-4*I+2*sqrt(3)+(2*I)*sqrt(3))^6);

evalf(u);

evalf[100](u): evalf(%);

simplify(sqrt(u));
    0

Yes, because the integral over 0..infinity is computed directly. Actually a human would do the same.

@Axel Vogt

f:=ln(x)^n*x^n/factorial(n);

int(f, x=1..x) assuming x>0;

(actually type int seems to be unrelated to integer)

@sarra 

So, you have an algebra and e[i] is a basis.

I think that a better approach would be to represent an element of the algebra sum( a[i]*e[i], i=0..n) by the vector <a[0],...,a[n]>, a[i] being the scalars.

Now, e[i] &x e[j] is also such a vector, but it seems that you do not know its  representation for all i,j. For example, what is e[2] &x e[3]  in terms of e[0],...,e[n]? It will be complicated to work in an algebra which is only partially defined.

@Rouben Rostamian  

I think I understand what OP wants:

1) e0, e1, ... are symbolic and are seen as a basis in an algebra (e.g. the quaternions)

2) * is a binary operator. In Maple it could be implemented as neutral; let's denote it by &x
It is actually the multiplicative operation in the algebra.

3) The operator  &x is bilinear and satisfies some relations, e,g. e0 &x e0 = e2

4) Using the bilinearity and these relations, expressions such as 
(e0 + 4*e3 - 2*e7) &x (-e0 + 2*e3 - 3*e4)
can be expanded and simplified using e.g. applyrule by giving the relations

5) f is a linear functional on the algebra and must be declared as such.
It is uniquely determined by f(e0), f(e1),...

Now, all OP's expressions have to be expanded and simplified.

@Bendesarts 

But in this case:
- the first answer works as well and it's shorter
- "periodic" in the title is not justified.