We know that the sum of any two sides of a triangle is always greater than the third side, in △OAB, we have,
OA + OB > AB ….. (i)
OB + OC > BC …… (ii)
OA + OC > CA ….. (iii)
On adding equations (i), (ii) and (iii) we get:
OA + OB + OB + OC + OA + OC > AB + BC + CA
2(OA + OB + OC) > AB + BC + CA
OA + OB + OC > (AB + BC + CA)/2
Or
OA + OB + OC > ½ (AB + BC +CA)
Hence the proof.