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 ;)

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TopNewestWell, 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 · 1 year, 5 months ago

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– Atanu Ghosh · 1 year, 4 months ago

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..Log in to reply

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 · 1 year, 4 months ago

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The heart of the problem is use of \(apollonius ..theorem\) over many times. – Shivam Jadhav · 1 year, 7 months ago

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