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.
\[\frac{1}{2}P < MA + MB + MC < P\]
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
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
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.

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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
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

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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.
\[\sum_{k=0}^{n1}e^{\frac{2 \pi i k}{n}}=0\]
Using $L^AT_EX$ and following the rules of the forum are very easy but really important, too.Please cooperate.
Using $L^AT_EX$ and following the rules of the forum are very easy but really important, too.Please cooperate.

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Re: Problems Involving Triangles
Number 2. \[13/2,5\]
@tiham, rookie mistake
@tiham, rookie mistake

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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.
By definition, Reflection of P across the midpoint of BC lies on AP. So all those reflective lines are concurrent at P.
Re: Problems Involving Triangles
Think again (and maybe draw a figure.)Ashfaq Uday wrote: Reflection of P across the midpoint of BC lies on AP
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Nur Muhammad Shafiullah  Mahi