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Question: If \(\bar{S}\) is the circumcentre, G is the centroid and O is the orthocentre of a ∆ABC then \(\ove...

If Sˉ\bar{S} is the circumcentre, G is the centroid and O is the orthocentre of a ∆ABC then SA____+SB____+SC____=\overset{\text{\_\_\_\_}}{\text{SA}} + \overset{\text{\_\_\_\_}}{\text{SB}} + \overset{\text{\_\_\_\_}}{\text{SC}} =

A

SG____\overset{\text{\_\_\_\_}}{\text{SG}}

B

OS____\overset{\text{\_\_\_\_}}{\text{OS}}

C

SO____\overset{\text{\_\_\_\_}}{\text{SO}}

D

OG____\overset{\text{\_\_\_\_}}{\text{OG}}

Answer

SO____\overset{\text{\_\_\_\_}}{\text{SO}}

Explanation

Solution

SA____+SB____+SC____=SA____+(SD____+DB____)+(SD____+DC____)\overset{\text{\_\_\_\_}}{\text{SA}} + \overset{\text{\_\_\_\_}}{\text{SB}} + \overset{\text{\_\_\_\_}}{\text{SC}} = \overset{\text{\_\_\_\_}}{\text{SA}} + \left( \overset{\text{\_\_\_\_}}{\text{SD}} + \overset{\text{\_\_\_\_}}{\text{DB}} \right) + \left( \overset{\text{\_\_\_\_}}{\text{SD}} + \overset{\text{\_\_\_\_}}{\text{DC}} \right)

= SA____+2SD____\overset{\text{\_\_\_\_}}{\text{SA}} + 2\overset{\text{\_\_\_\_}}{\text{SD}} (∵BD____=DC____\because\overset{\text{\_\_\_\_}}{\text{BD}} = \overset{\text{\_\_\_\_}}{\text{DC}}

= SA____+AO____\overset{\text{\_\_\_\_}}{\text{SA}} + \overset{\text{\_\_\_\_}}{\text{AO}} = SO____\overset{\text{\_\_\_\_}}{\text{SO}}