Orthocenter in Complex Plane

What is this about?

21 January 2016, Created with GeoGebra

Problem

Orthocenter in Complex Plane, problem

Proof

We know that, for a triangle with the circumcenter at the origin, the sum of the vertices coincides with the orthocenter. The orthocenter is known to fall outside the triangle if the triangle is obtuse. In this case, the orthocenter lies in the vertical pair of the obtuse angle:

Orthocenter in Complex Plane, obtuse case

It's thus clear that it also falls outside the circumcircle.

Acknowledgment

The idea of this page came up in a discussion with Leo Giugiuc of another problem.

 

Complex Numbers

  1. Algebraic Structure of Complex Numbers
  2. Division of Complex Numbers
  3. Useful Identities Among Complex Numbers
  4. Useful Inequalities Among Complex Numbers
  5. Trigonometric Form of Complex Numbers
  6. Real and Complex Products of Complex Numbers
  7. Complex Numbers and Geometry
  8. Plane Isometries As Complex Functions
  9. Remarks on the History of Complex Numbers
  10. Complex Numbers: an Interactive Gizmo
  11. Cartesian Coordinate System
  12. Fundamental Theorem of Algebra
  13. Complex Number To a Complex Power May Be Real
  14. One can't compare two complex numbers
  15. Riemann Sphere and Möbius Transformation
  16. Problems

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