Problems Involving Triangles

[willpower]

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.

[willpower]

Could anyone please post proofs to the above statements? Any help would be appreciated.

[Hasib]

Hey, i am in a journey. I'll try it at home after reaching Thanks for the message!

[Ashfaq Uday]

\[MA+MB> AB ,MB+MC>BC, MC+MA>CA\]
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]

opps. let M be any arbitrary point INSIDE the triangle

[nafistiham]

why don't you edit the post? it is better i think than mentioning it in a later post.

[Ashfaq Uday]

Number 2. \[13/2,5\]
@tiham, rookie mistake

[Ashfaq Uday]

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*]

Reflection of P across the midpoint of BC lies on AP

Think again (and maybe draw a figure.)