## Problems Involving Triangles

willpower
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Problems Involving Triangles
1. Denote by P the perimeter of triangle ABC. If M is a point in the interior of the triangle, prove that:
$\frac{1}{2}P < MA + MB + MC < P$

2. The side length of the equilateral triangle ABC equals l. The point P lies in the interior of ABC and the distances from P to the triangle’s sides are 1, 2, 3. Find the possible values of l.

3. Let P be a point in the interior of the triangle ABC. The reﬂections of P across the midpoints of the sides BC, CA, AB, are$P_{a}, P_{b}, P_{c}$ respectively. Prove that the lines APA, BPB, and CPC are concurrent.
Everybody is a genius; but if you judge a fish on its ability to climb a tree, it will live its entire life believing that it is stupid. - Albert Einstein

willpower
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Location:Pakistan

### Re: Problems Involving Triangles

Could anyone please post proofs to the above statements? Any help would be appreciated.
Everybody is a genius; but if you judge a fish on its ability to climb a tree, it will live its entire life believing that it is stupid. - Albert Einstein

Hasib
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### Re: Problems Involving Triangles

Hey, i am in a journey. I'll try it at home after reaching Thanks for the message!
A man is not finished when he's defeated, he's finished when he quits.

Ashfaq Uday
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Joined:Tue Sep 27, 2011 12:18 am

### Re: Problems Involving Triangles

$MA+MB> AB ,MB+MC>BC, MC+MA>CA$
Add the three inequalities and get the desired result.
Let $M$
be any arbitrary point on $\Delta ABC$
notice $MA$
has to be smaller than any of the three sides of the triangle. The rest is ur's to solve

Ashfaq Uday
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Joined:Tue Sep 27, 2011 12:18 am

### Re: Problems Involving Triangles

opps. let M be any arbitrary point INSIDE the triangle

nafistiham
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### Re: Problems Involving Triangles

why don't you edit the post? it is better i think than mentioning it in a later post.
$\sum_{k=0}^{n-1}e^{\frac{2 \pi i k}{n}}=0$
Using $L^AT_EX$ and following the rules of the forum are very easy but really important, too.Please co-operate.

Ashfaq Uday
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Joined:Tue Sep 27, 2011 12:18 am

### Re: Problems Involving Triangles

Number 2. $13/2,5$
@tiham, rookie mistake

Ashfaq Uday
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Joined:Tue Sep 27, 2011 12:18 am

### Re: Problems Involving Triangles

number 3. $AP,BP,CP$ are concurrent at $P$
By definition, Reflection of P across the midpoint of BC lies on AP. So all those reflective lines are concurrent at P.

*Mahi*
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### Re: Problems Involving Triangles

Ashfaq Uday wrote: Reflection of P across the midpoint of BC lies on AP
Think again (and maybe draw a figure.)
Use $L^AT_EX$, It makes our work a lot easier!