How do I plot this vectorial system in Cartesian coordinates regarding the triangle $ABC$?

Clear[a, b, c, i, j, g]
SeedRandom[1542]

Set up your equations and solve them:

equations = {
 2 (Array[i, 2] - Array[a, 2]) + (Array[i, 2] - Array[b, 2]) == 0,
   (Array[j, 2] - Array[b, 2]) - 2 (Array[j, 2] - Array[c, 2]) == 0,
 2 (Array[g, 2] - Array[a, 2]) + (Array[g, 2] - Array[b, 2]) - 2 (Array[g, 2] - Array[c, 2]) == 0
} // Simplify;

solutions = ToRules@Reduce[equations, Evaluate@Flatten@{Array[#, 2] & /@ {a, b, c, i, j, g}}];

Choose some random values for $A,B,C$ and calculate the corresponding $I,J,G$ using the rules obtained above:

Evaluate[Array[#, 2] & /@ {a, b, c}] = RandomReal[1, {3, 2}];
Array[#, 2] & /@ {i, j, g} /. solutions;

Create a graphical representation:

Show[
  (* plot points *)
  ListPlot[
    MapThread[
      {Labeled[Array[#1, 2], Style[#2, 16]]} &,
      {
        {a, b, c, i, j, g},
        {"a", "b", "c", "i", "j", "g"}
      }
    ] /. solutions,
    PlotStyle -> PointSize[0.02]
  ],

  (* add triangle and lines *)
  Graphics[{
    EdgeForm[{Thick, Black}], FaceForm[None],
    Triangle[Array[#, 2] & /@ {a, b, c}],
    {
      (*IA, IB*)
      Red,  Line[Array[#, 2] & /@ {i, a}], 
      Blue, Line[Array[#, 2] & /@ {i, b}],

      Purple,(*JB, JC*)
      Line[Array[#, 2] & /@ {j, b}], Line[Array[#, 2] & /@ {j, c}],

      Darker@Green,(*GA, GB, GC*)
      Line[Array[#, 2] & /@ {g, a}], 
      Line[Array[#, 2] & /@ {g, b}], 
      Line[Array[#, 2] & /@ {g, c}]
    } /. solutions
  }],
  PlotRange -> All
]

Use RandomInstance and GeometricScene to draw the picture:

ri = RandomInstance[
  GeometricScene[
   {
    {a, b, c, i, j, g},
    {a1, a2, b1, b2, c1, c2, i1, i2, j1, j2, g1, g2}
    },
   {
    a == {a1, a2}, b == {b1, b2}, c == {c1, c2}, i == {i1, i2}, 
    j == {j1, j2}, g == {g1, g2},
    Triangle[{a, b, c}],
    2*({a1, a2} - {i1, i2}) + ({b1, b2} - {i1, i2}) == {0, 0},
    ({b1, b2} - {j1, j2}) - 2*({c1, c2} - {j1, j2}) == {0, 0},
    2*({a1, a2} - {g1, g2}) + ({b1, b2} - {g1, g2}) - 
      2*({c1, c2} - {g1, g2}) == {0, 0}
    }
   ]
  ]

Extract the point coordinates:

ri["Points"]