# If O be a inner point of a triangle ABC Then prove that OA+OB+OC < AB+BC+CA ?

Jul 6, 2018

Let O be a inner point of a triangle ABC. We are to prove that $O A + O B + O C < A B + B C + C A$

*Construction"

$B O$ is produced to intersect AB at D.

For $\Delta A B D$

$A B + A D > B D$

$\implies A B + A D > B O + O D \ldots . . \left[1\right]$

For $\Delta C O D$

$C D + O D > O C \ldots \ldots \left[2\right]$

Adding [1] and [2] we get

$A B + A D + C D + O D > B O + O D + O C$

$\implies A B + A C + \cancel{O D} > B O + \cancel{O D} + O C$

$\textcolor{red}{\implies A B + A C > O B + O C \ldots . . \left(3\right)}$

Similarly

$\textcolor{b l u e}{A B + B C > O A + O C \ldots . . \left(4\right)}$

And

$\textcolor{g r e e n}{A C + B C > O A + O B \ldots . . \left(5\right)}$

$2 \left(A C + B C + C A\right) > 2 \left(O A + O B + O C\right)$
$\implies O A + O B + O C < A B + B C + C A$