Geometry Question - Complex numbers & triangles

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SUMMARY

The discussion focuses on finding complex numbers that represent the vertices of triangle ABC, given the midpoints of its sides as complex numbers z_1, z_2, and z_3. The user initially considers placing the origin at vertex A, leading to the vertices being expressed as A = O, B = 2*z_3, and C = 2*z_2. However, it is concluded that the choice of origin does not affect the solution, as the problem relates to the properties of similar triangles.

PREREQUISITES
  • Understanding of complex numbers and their geometric representation
  • Knowledge of triangle properties, specifically midpoints and similar triangles
  • Familiarity with coordinate systems and transformations
  • Basic algebraic manipulation of complex expressions
NEXT STEPS
  • Explore the concept of complex number transformations in geometry
  • Study the properties of similar triangles and their applications in complex analysis
  • Learn about the geometric interpretation of midpoints in the complex plane
  • Investigate the use of complex numbers in solving geometric problems
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Mathematicians, students studying complex analysis, and anyone interested in the geometric applications of complex numbers.

MattL
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OK, I've got this question to do:

Find complex numbers representing the vertices of a triangle ABC given
that the midpoints of the sides BC, CA, AB are represented by complex numbers
z_1, z_2, z_3 respectively.

Thing is, I don't know where I'm taking the origin to be; if I took it at A then I would just have A is O, B is 2*z_3, C is 2*z_2. Surely there's more to the question than that?
 
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yeah, oops, just realized it's about similar triangle and it doesn't matter where the origin is...
 

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