Question

O is a point in the exterior of ABC, then prove that
OA + OB + OC > 1/2 (AB + BC + CA)
​

Answer 1

Given -

O is a point in the exterior of ABC.

To prove -

OA + OB + OC > ½ (AB + BC + CA)

Construction -

Join OA, OB and OC

Property Used -

Sum of two sides in a triangle is greater than the third side .

Proof -

Using the above mentioned property - .

AO + OC > AC

AO + OB > AB

OB + OC > BC

Adding these three Inequalities -

2AO + 2BO + 2CO > AB + BC + AC

2 ( OA + OB + OC ) > ( AB + BC + AC )

( OA + OB + OC ) > ½ ( AB + BC + AC )

Hence proved

___________

Additional information -

1. Sum of two sides of a triangle is always greater than the third side .

2. Difference of two sides of a triangle is always less than the third side .

__________________