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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
2 years, 1 month 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 ).

- 1 year, 11 months ago

- 1 year, 10 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.

- 1 year, 10 months ago

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

- 2 years, 1 month ago