So, what exactly do you want? As you wrote, satisfying all four equations seems unreachable. And even more, I am quite sure that one can prove that the requested point does not exist using Mathematica built-ins:
ClearAll[x, y, n, points, eqns]
points = {{0, 0}, {20, 0}, {138/5, (24 Sqrt[6])/5}, {25, 10 Sqrt[6]}};
eqns = And @@ Table[Norm[{x, y} - point] == n, {point, points}];
Solve[
eqns && And[Element[x, Reals], Element[y, Reals], Element[n, Integers]],
{x, y, n}
]
These lines return an empty list which means that there's no solution with the required properties. Relaxing the constraint for n
yields a unique solution:
Solve[
eqns && And[Element[x, Reals], Element[y, Reals], Element[n, Reals]],
{x, y, n}
]
The result is a list {{x -> 10, y -> 145/(4 Sqrt[6]), n -> 175/(4 Sqrt[6])}}
. It does not sound probable that there's a second, "integer" solution to this problem which was not detected by the Solve
function.
So, is it what you are looking for?