find a point within a triangle such that the sum of the squares of its distances from three vertices is minimum

Question

find a point within a triangle such that the sum of the squares of its distances from three vertices is minimum

Answer 1

Assume the points are (x1,y1) (x2,y2) (x3,y3)
The point to be determined is (x,y)
F=[(x-x1)^2+(y-y1)^2]+
[(x-x2)^2+(y-y2)^2]+
[(x-x3)^2+(y-y3)^2]

dF/dx=2[x-x1]+2[x-x2]+2[x-x3]=0

 

This will give x=[x1 +x2+x3]/3

 

dF/dx=2[y-y1]+2[y-y2]+2[y-y3]=0

 

This will give y=[y1 +y2+y3]/3