21. O is any point in the interior of ΔABC. Prove that
(i) AB + AC > OB + OC
(ii) AB + BC + CA > OA + OB + OC
(iii) OA + OB + OC >1/2(AB+BC+CA)
Share
Sign Up to our social questions and Answers Engine to ask questions, answer people’s questions, and connect with other people.
Login to our social questions & Answers Engine to ask questions answer people’s questions & connect with other people.
(iii) Now, consider the triangles OAC, OBA and OBC.
We have:
OA+OC>ACOA+OB>ABOB+OC>BCAdding the above three equations, we get:OA+OC+OA+OB+OB+OC>AB+AC+BC⇒2(OA+OB+OC)>AB+AC+BCThus, OA+OB+OC>12(AB+BC+CA)