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In an equilateral triangle, prove tha...

In an equilateral triangle, prove that three times the square of one side is equal to four times the square of one of its altitudes.

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Let AB=BC=CA=a
In `triangle ADB and triangle ADC` .
(common) ,
`triangleADB cong triangleADC` ( R.H.S)
` BD = DC = (BC)/2 = a/2`
Now in `triangleADB` by pythagoras theorem
`AB^(2) = AD^(2) + BD^(2)`
`a^(2)= AD^(2)+ (a/2)^(2)`
` AD^(2) =a^(2)- (a^(2))/4 Rightarrow AD^(2) = (3a^(2))/4`
` 4AD^(2) = 3a^(2) Rightarrow 4AD^(2)=3AB^(2)`
Hence , 3 times the square of side is equal to 4 times the square of altitude.
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