If x1,y1,z1,x2,y2,z2 and x3,y3,z3 are the vertices of an equilateral triangle such that x1−22+y1−32+z1−42=x2−22+y2−32+z2−42=x3−22+y3−32+z3−42  then ∑x1+2∑y1+3∑z1

Clearly 2,3,4 is the circumcenter of the triangle.⇒2,3,4 is the centroid of the triangle.  (since the triangle is equilateral)⇒x1+x2+x33,y1+y2+y33,z1+z2+z33=2,3,4 ⇒x1+x2+x3=6, y1+y2+y3=9, z1+z2+z3=12 ⇒∑x1+2∑y1+3∑z1=60