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Monotonically Increasing Functions

In Δ ABC, D ...

Question

In $Δ A B C$ , D is any point inside BC. Then AB + BC + AC > 2AD
State whether the above statement is true or false.

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Solution

In $△ A B D$ , we have sum of two sides > third side
$=> A B + B D > A D$ -- (1)
Also, in $△ A D C$ , we have sum of two sides > third side
$=> A C + C D > A D$ ---- (2)
Adding equations $1, 2$ we get
$=> A B + (B D + C D) + A C > 2 A D$
$=> A B + B C + A C > 2 A D$

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