Complex numbers, plane and geometry

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SUMMARY

The discussion centers on proving the equivalence of three propositions involving complex numbers a, b, and c in the complex plane. The propositions state that an equilateral triangle ABC (T1) is formed, that j or j² is a solution to the quadratic equation az² + bz + c = 0, and that the relation a² + b² + c² = ab + bc + ca holds. The participant attempts to connect these propositions using properties of equilateral triangles and the quadratic formula, while clarifying the use of the imaginary unit, distinguishing between j and i.

PREREQUISITES
  • Understanding of complex numbers and their representation in the complex plane
  • Familiarity with the properties of equilateral triangles
  • Knowledge of quadratic equations and the quadratic formula
  • Clarification of the imaginary unit, specifically the distinction between j and i
NEXT STEPS
  • Study the properties of equilateral triangles in the context of complex geometry
  • Learn about the quadratic formula and its applications in complex number equations
  • Explore the implications of using different imaginary units (j vs. i) in mathematical contexts
  • Investigate geometric interpretations of complex number equations
USEFUL FOR

Mathematics students, particularly those studying complex analysis and geometry, as well as educators seeking to clarify concepts related to complex numbers and their geometric representations.

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Homework Statement


There are three complex numbers a, b and c. Show that these propositions are equals.
1. ABC (triangle from the three points in complex plane) is equilateral (T1).
2. j or j2 is the solution for az2 + bz + c = 0.
3. a2 + b2 + c2 = ab + bc + ca


Homework Equations


There is a hint. Equilateral triangles made from the bases AB, BC, and CA have centres of gravity from which we can construct another equilateral triangle (T2).


The Attempt at a Solution


T1 and T2 are equal triangles. They have the same heights and sides. I've tried to use the equation to solve quadratic equation (quadratic formula) and assumed j is a solution, hence j2 is compliment of j or [tex]\bar{j}[/tex]. I found ac-3b=1. I have no idea how to use the equation. Is my assumption correct? Or my approach to the question is wrong?
 
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Is this an engineering class? That is, is "j" the imaginary unit, j2= -1?
 
This is a mathematic class and we use i as the imaginary unit. I don't think that the teacher mistyped it.
 

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