# 3.  AM is a median of a triangle ABC.             Is $\small AB+BC+CA> 2AM$ ? (Consider the sides of triangles $\small \Delta ABM$ and $\small \Delta AMC$.)

P Pankaj Sanodiya

As we know that the sum of two sides of ANY triangle is always greater than the third side(Triangles Inequality Rule).

So,

In $\small \Delta ABM$ :

$\overline {AB}+\overline {BM}>\overline{AM}...........(1)$

In $\small \Delta AMC$ :

$\overline {AC}+\overline {CM}>\overline{AM}...........(2)$

Adding (1) and (2), we get

$\overline{AB}+\overline {AC}+\overline{BM}+\overline {CM}>\overline{AM}+\overline{AM}$

As we can see M is the point in line BC So, we can say

$\overline{BM}+\overline {CM}=\overline {BC}$

So our equation becomes

$\overline{AB}+\overline {AC}+\left (\overline{BM}+\overline {CM} \right )>\overline{AM}+\overline{AM}$

$\small AB+BC+CA> 2AM$.

Hence it is a True statement.

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