For the variable triangle ABC with fixed vertex C(1,2) and A,B having coordinates(cost,sint),(sint,-cost),find the locus of the centroid.

centroid of triangle  = [ x₁+x₂+x₃/3 , y₁+y₂+y₃/3 ]

centroid = [ (1+cost+sint)/3 , (2+sint -cost)/3 ] = (x,y)             ..................1

from eq 1

3x = 1 + cost + sint 

(3x-1) = sint + cost           ..................2

3y = 2 + sint - cost      

(3y-2) = sint - cost           ............... 3

now applying , (eq 2)²  + (eq3 )²

(3x-1)² + (3y-2)²  = 2

this is equation of circle

Approved