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Question
Let ABC be an acute scalene triangle, O and H be its circumcentre and orthocentre respectively. Further, let N be the mid point of OH. The value of vector from −−→NA+−−→NB+−−→NC is?
Let ABC be an acute scalene triangle, O and H be its circumcentre and orthocentre respectively. Further, let N be the mid point of OH. The value of vector from −−→NA+−−→NB+−−→NC is?
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Solution
The correct option is C 12−−→HO
−−→HE=−−→NE−−−→NH⇒12−−→AH=12[−−→NB+−−→NC]−12−−→OH(HE⊥BC,AH⊥BC)⇒−−→HE∥−−→AHHE=Rcos(A),AH=2Rcos(A)⇒−−→HE=12−−→AHEmidpointofBC⇒−−→NE=12[−−→NB+−−→NC]NmidpointofOH⇒−−→NH=12−−→OH⇒−−→NH−−−→NA=−−→NB+−−→NC+−−−→OH)⇒−−→NA+−−→NB+−−→NC=−−→HO+12−−→OH=12−−→HO
−−→HE=−−→NE−−−→NH⇒12−−→AH=12[−−→NB+−−→NC]−12−−→OH(HE⊥BC,AH⊥BC)⇒−−→HE∥−−→AHHE=Rcos(A),AH=2Rcos(A)⇒−−→HE=12−−→AHEmidpointofBC⇒−−→NE=12[−−→NB+−−→NC]NmidpointofOH⇒−−→NH=12−−→OH⇒−−→NH−−−→NA=−−→NB+−−→NC+−−−→OH)⇒−−→NA+−−→NB+−−→NC=−−→HO+12−−→OH=12−−→HO
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