you gonna proof................

Geometry Level 2

In any Triangle, four times the sum of the squares of the medians will be equal to which if the following options?

Please don't google this you will lose fun.........

5/2 times the sum of squares of its sides 3 times the sum of squares of its sides 2 times the sum of its sides 4 times the sum of its sides

Rohan Kumar by Rohan Kumar , 22, India

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3 solutions

Anukriti Singh
Apr 14, 2014

this could be solved by using appollonius theorem !

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Moshiur Mission
Mar 13, 2014

AB, BC & CA are the any Triangle and AD, BE & CF are three Median. for simplicity we take <A=<B=<C=60 degree. so tan30 = DC/AD or 1/root3 = 1/2 BC/AD as BC = AB then root3AB = 2 AD or 3AB^2 = 4AD^2 similarly we get 3BC^2 = 4BE^2 and 3AC^2 = 4 CF^2 hence 3(AB^2 + BC^2 + AC^2) = 4 (AD^2 + BE^2 + CF^2)

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Yash Bansal
Mar 11, 2014

Simplest approach, which I took, was to consider an _equilateral triangle of unit sides _ .

Square of Length of median = (1 x sin60)^2 = 3/4

4 x (sum of squares of 3 medians) =4 x (3 x 3/4) = 9

Sum of squares of sides = 3

Hence, ratio is 3.

The correct proof can be done by the general method of co-ordinate geometry.

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