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 ;)

#Geometry

Easy Math Editor

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Comments

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 ).

Venkata Karthik Bandaru - 5 years, 3 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..

Atanu Ghosh - 5 years, 3 months ago

The heart of the problem is use of $apollonius ..theorem$ over many times.

Shivam Jadhav - 5 years, 6 months ago

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 - 5 years, 3 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$