A simple problem related with the following triangle

Question

I am stuck with the following problem that says :

$O$ is any point inside $\triangle ABC$. Then I have to prove that

1. $AB+AC \gt OB+OC$
2. $AB+BC+CA \gt OA+OB+OC$

MY TRY:

1. From $\triangle ABC$,$AB+AC \gt BC$ and from $\triangle OBC\,\,$,we get $\,\,OB+OC \gt BC$. Now ,I am not sure about how to proceed.

2. Considering the $\triangle AOB$,$\triangle BOC $,$\triangle AOC$, we get

   $OA+OB \gt AB\,\,,OB+OC \gt BC\,\,,OC+OA \gt AC\,\, $ respectively.

Now, combining all we get $2(OA+OB+OC) \gt (AB+BC+CA)$.

So,I am nowhere near the proof. Can someone point me in the right direction? Thanks in advance for your time.

Answer 1

For 1. :

Let $E$ be a point both on $OB$ and on $AC$.

From $\triangle{ABE}$, $$AB+AE\gt BO+OE\tag1$$

From $\triangle{OEC}$, $$OE+EC\gt OC\tag2$$

Now the claim follows from $(1)+(2)$.

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For 2. :

You can use 1. to prove that the inequality holds.