What is the relationship between the incenter and orthocenter of a triangle?

  • Context: MHB 
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Discussion Overview

The discussion centers around the geometric relationship between the incenter and orthocenter of a triangle, specifically exploring properties related to angle trisection and geometric proofs. The scope includes theoretical exploration and geometric reasoning.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant suggests that demonstrating the relationship between the incenter and orthocenter may also relate to the possibility of angle trisection using only a non-graduate ruler and compass.
  • Another participant asserts that while it is impossible to trisect any angle with only a non-graduate ruler and compass, some specific angles can be trisected, indicating a distinction that may not directly relate to the main problem.
  • A third participant seeks to prove that the angle between the extended line from point C to the orthocenter (CH) and the line from the orthocenter to the incenter (HI) measures half of angle B, requesting hints for a purely geometric proof.

Areas of Agreement / Disagreement

Participants express differing views on the relationship between angle trisection and the main problem, with no consensus on the implications of the incenter and orthocenter relationship. The discussion remains unresolved regarding the geometric proof sought by the third participant.

Contextual Notes

There are limitations regarding the assumptions about angle trisection and the definitions of the geometric points involved, which may affect the clarity of the relationships discussed.

Albert1
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What I see before 'attacking' the problem is that, without any serious failure of me, the 'demonstration' [if any...] would also demonstrate that trisection of an angle with only non graduate ruler and compass is possible... or not?... Kind regards $\chi$ $\sigma$
 
chisigma said:
trisection of an angle with only non graduate ruler and compass is possible... or not?...
It is impossible to trisect any angle with only non graduate ruler and compass ,but some angles are possible, and this is unrelated to this problem .
 
Last edited:
Albert said:
30kdffq.jpg
I need to show that the angle between $CH$(extended) and $HI$ measures $\frac{\angle B}{2}$. I am trying to give a purely geometric proof of this. Any hints on these lines?
 

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