Your plot can be constructed as follows:

X:=[seq(k, k=4..10, 0.1)]:

Y:=[]:

for k from 1 to 61 do

solve(eval(sqrt(x^2+y^2)=sqrt((x-4)^2+(y-3)^2)+5, x=X[k])):

Y:=[op(Y), [X[k], %]]:

od:

plot(Y, thickness=3, view=[-1..10, -1..8]);

1) To Alejandro Jakubi.

Thanks for the detailed explanation! It is unfortunate that Maple unable to cope with checking obvious statements.  

Another example:

is(2*х^5 +1>0) assuming x>=0, x<=1;

                     false

This is just a bug!

2) To Markiyan Hirnyk.

I think that the use of  continuous=true option, without proof of the convergence of the improper integral, is unacceptable because may result in an error.

An example:

int(1/x, x=-1..2, continuous=true);

             -Pi*I+ln(2)

Received an incorrect result!

1) To Alejandro Jakubi.

Thanks for the detailed explanation! It is unfortunate that Maple unable to cope with checking obvious statements.  

Another example:

is(2*х^5 +1>0) assuming x>=0, x<=1;

                     false

This is just a bug!

2) To Markiyan Hirnyk.

I think that the use of  continuous=true option, without proof of the convergence of the improper integral, is unacceptable because may result in an error.

An example:

int(1/x, x=-1..2, continuous=true);

             -Pi*I+ln(2)

Received an incorrect result!

@Markiyan Hirnyk 

1) Why was the combinat package called, if in the code it is not used?  

2) Your procedure s does not work in the classic interface, as left single quotes should be used: print(`Invalid input`) . In standard interface s works - Maple interprets a blank as multiplication.    

3) A simple exhaustive search program more effective (running at least 10 times faster)

s1:=proc(L::list)

local n, N, i, j;

n:=nops(L);

if nops(convert(L,set))=n then

N:=0;

for i to n-1 do

for j from i+1 to n do

if L[i]>L[j] then N:=N+1; fi;

od; od;

else print(`Invalid input`); fi;

(-1)^N;

end proc;

Examples:

L:=combinat[randperm](1000):

st:=time():

s1(L), time()-st;

    1,  0.967

st:=time():

s(L), time()-st;

    1,  11.981

@Markiyan Hirnyk 

1) Why was the combinat package called, if in the code it is not used?  

2) Your procedure s does not work in the classic interface, as left single quotes should be used: print(`Invalid input`) . In standard interface s works - Maple interprets a blank as multiplication.    

3) A simple exhaustive search program more effective (running at least 10 times faster)

s1:=proc(L::list)

local n, N, i, j;

n:=nops(L);

if nops(convert(L,set))=n then

N:=0;

for i to n-1 do

for j from i+1 to n do

if L[i]>L[j] then N:=N+1; fi;

od; od;

else print(`Invalid input`); fi;

(-1)^N;

end proc;

Examples:

L:=combinat[randperm](1000):

st:=time():

s1(L), time()-st;

    1,  0.967

st:=time():

s(L), time()-st;

    1,  11.981

s([1, 2, 3, 4]);

             -1

No comments!

s([1, 2, 3, 4]);

             -1

No comments!

Many thanks to all for the detailed comments and answers!

Mathematica immediately finds the correct answer:

N[FractionalPart[10^9*Exp[1]], 10]

0.4590452354

The coordinates of T and E are the rational numbers, what you would like!

The coordinates of T and E are the rational numbers, what you would like!

To toandhsp! I solved your problem by Classic Worksheet Maple 13  (time about 6 min).

To Markiyan Hirnyk! Your solving found only 98 solutions (and accurate of them will be 50). Therefore, I believe that the application of DirectSearch package for these tasks is inefficient. It is interesting your comment on this.

To toandhsp! I solved your problem by Classic Worksheet Maple 13  (time about 6 min).

To Markiyan Hirnyk! Your solving found only 98 solutions (and accurate of them will be 50). Therefore, I believe that the application of DirectSearch package for these tasks is inefficient. It is interesting your comment on this.

Dear Markiyan! Thank you for finding error in my code. I did not notice the coefficients 2, 3 and so on before x_2, x_3, and so on. The corrected code:

P:=proc(k, i)

local L, s, M, K, m;

if not (k>=2*i and 2*i>=4) then

error `should be k>=2*i>=4` fi;

if not type(k, posint) or not type(i, posint) then

error `should be k is integer and i is integer` fi;

L:=[];

for s from k-2*i to k-i do

M:=combinat[composition](s+i, i);

K:=[seq([seq(M[j,l]-1, l=1..i)], j=1..nops(M))];

for m from 1 to nops(M) do

if convert([seq(type(K[m,t]/t, integer), t=2..i)], `and`) then

L:=[op(L), [K[m,1],seq(K[m,t]/t, t=2..i)]]; fi; od;

od;

L;

end proc;

Dear Markiyan! Thank you for finding error in my code. I did not notice the coefficients 2, 3 and so on before x_2, x_3, and so on. The corrected code:

P:=proc(k, i)

local L, s, M, K, m;

if not (k>=2*i and 2*i>=4) then

error `should be k>=2*i>=4` fi;

if not type(k, posint) or not type(i, posint) then

error `should be k is integer and i is integer` fi;

L:=[];

for s from k-2*i to k-i do

M:=combinat[composition](s+i, i);

K:=[seq([seq(M[j,l]-1, l=1..i)], j=1..nops(M))];

for m from 1 to nops(M) do

if convert([seq(type(K[m,t]/t, integer), t=2..i)], `and`) then

L:=[op(L), [K[m,1],seq(K[m,t]/t, t=2..i)]]; fi; od;

od;

L;

end proc;