Question

In a triangle ABC if O is any point inside the triangle then prove that AB + BC + CA < 2 (AO + BO + CO ).

Answer 1

Join OA, OB and OC.
According to triangle inequality, sum of any two sides is greater than the third side.
Therefore,
OA+OB > AB ……(1)
OB+OC > BC ……(2)
OC+OA > AC ……(3)
Adding (1), (2) and (3), we get
2(OA+OB+OC) > (AB+BC+AC)
Dividing both sides by 2, we get
(OA+OB+OC) > 1/2(AB+BC+AC).

we get AB + BC + CA < 2 (AO + BO + CO ).