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.

 

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