MHB Proving $PA+PB+PC\geq MA+MB+MC$ in $\triangle ABC$

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In an acute triangle ABC, with M as an inner point where angles AMB, BMC, and CMA each measure 120 degrees, the inequality PA + PB + PC ≥ MA + MB + MC is to be proven for any point P within the triangle. The discussion emphasizes the geometric properties of the triangle and the implications of the angles formed at point M. Participants explore various approaches to demonstrate the inequality, focusing on the relationships between the distances from points A, B, C, and M to point P. The proof hinges on understanding the triangle's structure and the positioning of point P relative to M. Ultimately, the goal is to establish the validity of the inequality under the given conditions.
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M is an inner point of acute $\triangle ABC$

$\angle AMB=\angle BMC=\angle CMA=120^ o$

point P is another point in $\triangle ABC$

Prove :$PA+PB+PC\geq MA+MB+MC$
 
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Albert said:
M is an inner point of acute $\triangle ABC$

$\angle AMB=\angle BMC=\angle CMA=120^ o$

point P is another point in $\triangle ABC$

Prove :$PA+PB+PC\geq MA+MB+MC$
Consider a slightly different question.

Fix a number $l$.
Let $L$ be the locus of all the points $Q$ such that $|QB|+|QC|=l$.
Then $L$ is an ellipse.
Suppose we want to find a point $Q^*$ on $L$ such that $|AQ^*|+|Q^*B|+|Q^*C|$ is minimum.
Imagine a circle whose center is $A$ and whose radius expands with time. At time $t=0$ assume the radius of the circle is $0$.
At some point in time, say $t=t^*$, the circle first comes in contact with the ellipse $L$. Say the radius of this circle is $r^*$ and denote this circle as $C^*$.
One can show that $C^*\cap L$ is a singleton.
Say $C^*\cap L=\{Q^*\}$.
By this construction, we can also see that $Q^*$ is the point on $L$ such that $|AQ^*|+|Q^*B|+|Q^*C|$ is smallest.
By the properties of ellipse, we can also see that $AQ^*$ bisects angle $\angle BQ^*C$.

Now to our problem. Fermat point can be shown to exist in any acute angled triangle. If $F$ is the Fermat point, then $AF$ bisects angle $\angle BFC$, $BF$ bisects angle $\angle AFC$ and $CF$ bisects angle $\angle AFB$. By the above discussion, the inequality in the original question is easily established.
 
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Albert said:
M is an inner point of acute $\triangle ABC$

$\angle AMB=\angle BMC=\angle CMA=120^ o$

point P is another point in $\triangle ABC$

Prove :$PA+PB+PC\geq MA+MB+MC$
 

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