Question

in a triangle ABC ,O is an important point ,prove that 2(OA+ OB + OC) greater than AB+BC+CA .​

Answer 1

Given:-

• O is an external point for a triangle ABC.

To prove:-

• 2(OA+OB+OC)>AB+BC+CA

Construct:-

• Join OA, OB and OC.

Proof:-

We know that Sum of any two sides of a triangle is greater than the third side

From Δ OAB

OA + OB > AB ------------- (1)

Similarly,

From Δ OBC and Δ OAC

OB + OC > BC ------------- (2)

OA + OC > CA ------------- (3)

By adding equations (1), (2) and (3), we get

→OA + OB + OB + OC + OA + OC > AB + BC + CA

2OA+2OB+2OC

→2(OA + OB + OC) > (AB + BC + CA)

•°•Hence proved