Question related to triangles.

Question

I am stuck at a question:

O is a point in the interior of ∆PQR , then which of the following is true:

1)$(OP+OQ+OR)<1/2(PQ+QR+PR)$

2)$(OP+OQ+OR)=1/2(PQ+QR+PR)$

3)$(OP+OQ+OR)>1/2(PQ+QR+PR)$

Please can someone explain me how to do this?

Thanks for the help.

Answer 1

I will use the following fact to prove your result:

Fact: In a triangle sum of two sides is always greater than 3rd side

So $PQ~<~OP+OQ$

$PR~<~OP+OR$

$QR~<~OQ+OR$

Now adding these equations we have

$PQ+PR+QR<2(OP+OQ+OR)$

So $(OP+OQ+OR)>\frac{1}{2}(PQ+PR+QR)$