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

Let $$ABC$$ be a triangle from plane such that we define three points $$I$$, $$J$$ ,$$G$$ satisfying respectively the following equations: $$2 \vec{IA}+\vec{IB} =\vec{0}$$ $$\vec{JB}-2\vec{JC} =\vec{0}$$ $$2\vec{GA}+\vec{GB} -2 \vec{GC}=\vec{0}$$

How I can plot that system of vectorial equation in Cartesian coordinates with respect to triangle $$ABC$$? By hand, I obtained $$G$$ as a point of interesction between $$(CI)$$ and $$(AG)$$ such that $$(AC)$$ and $$(BG)$$ are parallel.

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[
{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"]