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

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The discussion centers on the geometric relationship between the incenter and orthocenter of a triangle, specifically regarding the angle between the extended line CH and HI, which is proposed to measure half of angle B. The conversation touches on the impossibility of trisecting an angle using only a non-graduated ruler and compass, although some specific angles can be trisected. The focus is on providing a geometric proof for the angle measurement in question. Participants are seeking hints and guidance to develop this proof. The exploration of these geometric properties highlights the complexities involved in triangle constructions.
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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 .
 
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Albert said:
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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?
 

Thread 'Trigonometry problem of interest'

Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
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