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- November 6th 2006, 08:47 PM #1

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## Question 5

**Im**PerfectHacker has requested that I set a problem this week so here it is:

Problem of the Week 5

Given any four distinct points , show that the three angles between the bisectors of , and are all acute, right or obtuse.

RonL

(Clarification: the bisectors are of the non-reflex angles that correspond to the specified points)

- November 7th 2006, 06:01 AM #2

- November 7th 2006, 07:44 AM #3

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- November 12th 2006, 09:23 PM #4

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Introduce unit vectors along and respectively. Then

the bisectors are collinear with and . Also:

So these three dot products are equal, so in particular are all positive,

zero or negative. Which implies that the angles between the bisectors

are all acute, right or obtuse.

RonL

- November 20th 2006, 03:55 PM #5

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A couple of questions before I try to solve it: (The previous solution used concepts I am unfamiliar with so the problem isn't ruined for me )

Can I assume A,B,C, and O are noncollinier?

Am I trying to prove that all of the angles have the same quality of acute, obtuse, and/or right, or only that none of them are reflexive or lines?

Also, how much geometry is required at mininum to find the proof? I've only taken 1 year.

- November 20th 2006, 08:39 PM #6

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You can assume so, I see an exeptional and/or ambiguous cases

if the can be colinear. What I actualy want is the O does not lie on

any of the segmants AB, BC, AC.

Am I trying to prove that all of the angles have the same quality of acute, obtuse, and/or right, or only that none of them are reflexive or lines?

Also, how much geometry is required at mininum to find the proof? I've only taken 1 year.

RonL