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

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Answer 1 · 2018-07-06T12:50:11

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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#