If the vertices of the triangles are ( 1 , 2 , 3 ) , ( 2 , 3 , 1 ) , ( 3 , 1 , 2 ) and if H ,   G ,…

If the vertices of the triangles are (1,2,3),(2,3,1),(3,1,2) and if H, G, S and I respectively denote its orthocenter, centroid, circumcenter and incenter, then H+G+S+I=
  1. (2,2,2)
  2. (4,4,4)
  3. (6,6,6)
  4. (8,8,8)

Solution

Distance between x,y,z & a,b,c is x-a2+y-b2+z-c2.

AB=1+1+4=6

BC=1+1+4=6

AC=4+1+1=6

So it is an equilateral triangle

H=S=I=G=1+2+33,2+3+13,3+1+23≡2,2,2

⇒H+S+I+G=2+2+2, 2+2+2, 2+2+2=8, 8, 8

Asked in: AP EAMCET 2021 (20 Aug Shift 1)

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