@Rouben Rostamian   min(a,b) = (a + b - abs(a - b))/2

@one man  Try in the procedure to replace  r:=sqrt(add(d^~2))  with  r:=sqrt(`+`((d^~2)[ ]))  or  r:=sqrt(add(d[i]^2, i=1..nops(d))) .  Do not forget to restart first.

@Harry Garst  Here are three simple examples that confirm your observation:

Expr1:=a/(a+b+c);
algsubs(a+b+c=d, Expr1);

Expr2:=b/(a+b+c);
algsubs(a+b+c=d, Expr2);

Expr3:=c/(a+b+c);
algsubs(a+b+c=d, Expr3);

We see that the result depends on the position:  a  is the first in  a+b+c

The code snippet that you submitted works as expected. Give the complete code with which you have problems.

my_proc:=proc(func::`+`) 
    subs([x[1] = 2, x[2] = 1], func);
end proc:

func:=5 + x[1]*x[2] + 10*x[1];
my_proc(func);

                                      func := x[1] x[2] + 10 x[1] + 5
                                                         27
 

@Rouben Rostamian  A brilliant solution, which is better than mine. Unfortunately, there are examples with which it fails, eg for  cos(x+y+1)+sin(x-1)*sin(y+2)

PS. It is worth noting that Mathematica copes with this:
Simplify[Cos[x + y + 1] + Sin[x - 1]*Sin[y + 2]]
                     Cos[1 - x] Cos[2 + y]

@nm  I adjusted the procedure a little. Now it works with exp function, but still does not work with sqrt:

Separation(exp(x^2-y));

                               [exp(x^2), exp(-y)]

@idol050279  I do not understand what relation CrossSectionTutor command has to the points inside the cylinder. This command simply shows the intersection lines of some surface with a set of planes.
Write more clearly what exactly you want to achieve.

@acer  But why  match  command fails with this example (in Maple 2017.1). Is this a bug?

restart;
e:= x^2+y^2-8*x-12*y-92:
g:= (x+a)^2+(y+b)^2+c:
match(e=g, {x,y}, 's');
s;
                            false
                               s

@Christopher2222  Change the ranges for the plotting:

restart;
F := x^2+y^2+a*x+b*y+c:
XY := [[8,0], [4,4], [14,4]]:
minimize(add((eval(F, {x = XY[i, 1], y = XY[i, 2]}))^2, i = 1 .. nops(XY)), location);
assign(op(%[2])[1]):
A := plot(XY, style = point, symbolsize = 15, symbol = solidcircle, color = red):
B := plots[implicitplot](F, x = -1 .. 15, y = -1 .. 11, color = green):
plots[display](A, B, scaling = constrained);

@ernilesh80  You wrote "can I also get  sign (positive or negative) of determinant of all possible 63 principal minors? "

Any principal minor is determined by some sublist of the list  [1,2, ..., n]  (A is a square matrix n x n). Here is a procedure that returns the positions and determinants of all leading principal minors (lpm) or principal minors (pm) of a matrix A:

Minors:=proc(A, p)
local n, L1, L2;
uses LinearAlgebra, combinat:
n:=Dimension(A)[1];
L1:=[seq([$1..k], k=1..n)];
if p=lpm then return seq(k=Determinant(A[k,k]), k=L1) else
L2:=subsop(1=NULL, powerset([$1..n]));
seq(k=Determinant(A[k,k]), k=L2) fi;
end proc:
 

Example of use:

A:=LinearAlgebra:-RandomMatrix(3, shape=symmetric);
Minors(A, lpm);  # Leading principal minors
Minors(A, pm);  # Principal minors

@mohkam7  A slight adjustment of the procedure allows you also to work with polynomials containing symbolic coefficients, provided that conditions are imposed on them:

ClassTerms1:=proc(P::polynom, S::list(symbol), Ass::set:={})
local C, t, T, Terms, Tp, Tn, Tind;
C:=[coeffs(P,S,'t')];
T:=[t];
Terms:=convert(zip(`[]`,C,T),set);
Tp:=select(p->is(p[1]>0) and `and`(seq(type(degree(p[2],s),even), s=S)), Terms) assuming op(Ass);
Tn:=select(p->is(p[1]<0) and `and`(seq(type(degree(p[2],s),even), s=S)), Terms minus Tp) assuming op(Ass);;
Tind:=Terms minus `union`(Tp,Tn);
print(P_positive={seq(p[1]*p[2],p=Tp)});
print(P_negative={seq(p[1]*p[2],p=Tn)});
print(P_indeterminate={seq(p[1]*p[2],p=Tind)});
end proc:

Example of use:

ClassTerms1(a*x^2+b*x*y-c*y^2+d*x^2*y^2+x-5, [x,y], {a<0,b>0,c<0,d<0});

@Jalale  You can use  Explore  command to create similar animations with sliders. See help on this command.

@Jalale  I did not understand what you mean by  "... animation with distances like AB/A'B' =AC/A'C'..."

@savalan I do not understand what you wrote at all. An equation does not depend on how you designate its roots. Your equations in the for loop are the same initial equations with the same roots, only the expression for the root  d  is shifted because you are actually making a change of variable in the same equations.

@savalan   What's the problem? Here are the first 50 positive values of  d:

seq(eval(rhs(sol[2]), _Z1=z), z=0..49);
0.03689590440, 0.5368959044, 1.036895904, 1.536895904, 2.036895904, 2.536895904, 3.036895904, 3.536895904, 4.036895904, 4.536895904, 5.036895904, 5.536895904, 6.036895904, 6.536895904, 7.036895904, 7.536895904, 8.036895904, 8.536895904, 9.036895904, 9.536895904, 10.03689590, 10.53689590, 11.03689590, 11.53689590, 12.03689590, 12.53689590, 13.03689590, 13.53689590, 14.03689590, 14.53689590, 15.03689590, 15.53689590, 16.03689590, 16.53689590, 17.03689590, 17.53689590, 18.03689590, 18.53689590, 19.03689590, 19.53689590, 20.03689590, 20.53689590, 21.03689590, 21.53689590, 22.03689590, 22.53689590, 23.03689590, 23.53689590, 24.03689590, 24.53689590