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Question

# If O is a point within ΔABC then show that :1) AB+AC=OB+OC2) AB+BC+CA>OA+OB+OC3) OA+OB+OC>12(AB+BC+CA).

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Solution

## (i) It is given that O is a point within △ABCConsider △ABCWe know that AB+AC>BC..(1)Consider △OBCWe know that OB+OC>BC..(2)By subtracting both the equations we get(AB+AC)(OB+OC)>BC−BCSo we get(AB+AC)(OB+OC)>0AB+AC>OB+OCTherefore, it is proved that AB+AC>OB+OC.(ii) We know that AB+AC>OB+OCIn the same way, we can writeAB+BC>OA+OC and AC+BC>OA+OBBy adding all the equations we getAB+AC+AB+BC+AC+BC>OB+OC+OA+OC+OA+OBSo we get2(AB+BC+AC)>2(OA+OB+OC)Dividing by 2 both sidesAB+BC+AC>OA+OB+OC(iii) Consider △OABWe know that OA+OB>AB..(1)Consider △OBCWe know that OB+OC>BC..(2)Consider △OCAOC+OA>CA(3)By adding all the equationsOA+OB+OB+OC+OC+OA>AB+BC+CAAB+BC+AC>OA+OB+OC(iii) Consider △OABWe know that OA+OB>AB..(1)Consider △OBCWe know that OB+OC>BC..(2)Consider △OCAOC+OA>CA(3)By adding all the equationsOA+OB+OB+OC+OC+OA>AB+BC+CA.OA+OB+OC>12(AB+BC+CA).

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