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"]