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 ∑$$x1+2$$∑$$y1+3$$∑$$z1$$

Question

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 ∑$$x1+2$$∑$$y1+3$$∑$$z1$$

Infinity Learn · Accepted Answer

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

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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

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