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I only want to get all the solutions of this system of the solutions with a value of $k$, we have the same values ​​of $m$

A = {1, 2, 3};
B = {2, 1, 0};
M = {a, b, c};
Solve[{a - b + c + m == 0, 
  SquaredEuclideanDistance[B, M] == k, (A - M). (B - M) == 0, 
  k > 0}, {a, b, c, m, k}, Integers]

In this ystem of the solutions, for example, $k = 6$, $m=1$ and $m = 3$.

A = {1, 2, 3};
B = {2, 1, 0};
M = {a, b, c};
Solve[{a - b + c + m == 0, 
  SquaredEuclideanDistance[B, M] == 6, (A - M). (B - M) == 0}, {a, b, 
  c, m}, Integers]

I do not how to start. How do I tell Mathematica to do that?

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2 Answers 2

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Perhaps

  {#, GatherBy[Solve[{a - b + c + m == 0, 
  SquaredEuclideanDistance[B, M] == #, (A - M).(B - M) == 0}, 
  {a, b, c, m}, Integers], Last]} & /@ Range[10];
  Grid[%]

enter image description here

or

 Table[{k, GatherBy[
   Solve[{a - b + c + m == 0,
   SquaredEuclideanDistance[B, M] == k,
   (A - M).(B - M) == 0},
   {a, b, c, m}, Integers], 
  Last]},
  {k, Range[10]}];
 Grid[%]

or, (the solutions grouped by the values of k and m?):

GatherBy[
Solve[{a - b + c + m == 0,
SquaredEuclideanDistance[B, M] == k,
(A - M).(B - M) == 0},
{a, b, c, k, m}, Integers], 
#[[-2 ;;]] &]
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By the linguistically esoteric phrase,

"...with a value of k, we have the same values ​​of m"

I guess you mean that you want to collect/filter the set of solutions given by Solve for cases where values of both $k$ and $m$ are constant! If this is indeed the case then try the following where res is the list of $23$ solution returned by your Solve.

collected=
Sort[
 Select[
   GatherBy[res, (#[[4]] && #[[5]]) &],
   Length@# >= 2 &
   ],
 #1[[1, 5, 2]] < #2[[1, 5, 2]] &
]

This gives us $10$ pairs of solutions where for a fixed value of $k$ we have same value of $m$. The Sort is used just to order the pairs according to the value of $k$. Following are the $3$ outliers that do not follow above pattern are

Complement[res, collected]

{{a -> 1, b -> 2, c -> 0, m -> 1, k -> 2},

{a -> 1, b -> 2, c -> 3, m -> -2, k -> 11},

{a -> 2, b -> 1, c -> 3, m -> -4, k -> 9}}

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