If O be a inner point of a triangle ABC Then prove that OA+OB+OC < AB+BC+CA ?

1 Answer
Jul 6, 2018

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Let O be a inner point of a triangle ABC. We are to prove that OA+OB+OC < AB+BC+CA

*Construction"

BO is produced to intersect AB at D.

For DeltaABD

AB+AD>BD

=>AB+AD>BO+OD.....[1]

For DeltaCOD

CD+OD>OC......[2]

Adding [1] and [2] we get

AB+AD+CD+OD>BO+OD+OC

=>AB+AC+cancel(OD)>BO+cancel(OD)+OC

color(red)(=>AB+AC>OB+OC.....(3))

Similarly

color(blue)(AB+BC>OA+OC.....(4))

And

color(green)(AC+BC>OA+OB.....(5))

Adding (3),(4)and (5) we get

2(AC+BC+CA)>2(OA+OB+OC)

=>OA+OB+OC < AB+BC+CA