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Wild problem

In \(\Delta ABC\) , let \(D,E,F\) be mid points of \(BC \ , \ AC \ , \ AB\) respectively and \(G\) be the centroid. Then prove that for all points \(K\) in the plane of triangle,

\[\large AK^2+BK^2+CK^2+9GK^2=4(DK^2+EK^2+FK^2)\]

Expecting numerous solutions.
Let's see who has the most elegant solution ;)

Note by Nihar Mahajan
11 months, 4 weeks ago

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Well, there is a co-ordinate bash solution, but I don't think it is worth to type out the whole thing ( since the bash is trivial to do ). Karthik Venkata · 9 months, 3 weeks ago

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@Karthik Venkata I heard about you, you are SHINING STAR of our future . Please read out my 3rd note which is available on my PROFILE please read it out. I AM SURE IT WILL DEFINITELY help you to achieve your GOAL BEST OF LUCK.. Atanu Ghosh · 9 months, 1 week ago

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Nihar Mahajan Sir; I have put a new topic as note on my profile. SCIENCE-a boon or bane PLEASE PUT YOUR POINT OF VIEWS ON THE DISCUSSION. Atanu Ghosh · 9 months, 1 week ago

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The heart of the problem is use of \(apollonius ..theorem\) over many times. Shivam Jadhav · 11 months, 3 weeks ago

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