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I have this question as homework:
Find the minimum value of the expression: $|z|^2 +|z-3|^2 +|z-6i|^2.$
Here's what I did: I plotted the points (0,0), (0,6), and (3,0) on the argand plane and joined them to make a triangle.
Now here is where I doubt myself. I found it centroid (1,2) and this should be the centre of mass if unit masses are kept at every vertex. Yes, centre of mass. Now the Moment of Inertia of a planar object is minimum about the axis passing perpendicular through its centre of mass and also from the parallel axis theorem $ I = Icom + md^2$ where $ Icom = (1)|z|^2 + (1)|z-3|^2 + (1)|z-6i|^2$ I is the moment of Inertia about any axis parallel to Icom at a distance d from it.
So the minimum value of expression must be minimum about this point only. My answer is also correct.
If I am correct how do to explain it mathematically or using mathematic theorems or axioms. Otherwise how do I do it.
I can shift this to Physics Stack Exchange if this doesn't fit here. Sorry for no LaTeX as I have just started using Stack Exchange from mobile. Please edit whatever required.