First slide
Introduction to 3-D Geometry
Question

 If x1,y1,z1,x2,y2,z2 and x3,y3,z3 are the vertices of an equilateral triangle such that 

x122+y132+z142=x222+y232+z242=x322+y332+z342  then x1+2y1+3z1

Moderate
Solution

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+2y1+3z1=60

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