Question

prove that three times the square of any side of an equilateral triangle is equal to four times the square of the altitude

Answer 1

consider the side of a triangle is a
and the median is AM
we know that AM=   a√3/2
4 times altitude =4(a√3/2)whole square
=4(3 asquare/4)
=3 asquare

Answer 2

Let the side of an equilateral triangle = a units

Length of the altitude of the triangle = $\frac{ \sqrt{3}a}{2}$

4 times the square of the altitude = $4( \frac{ \sqrt{3} a}{2} )^2$
  
                                                    = $4 * \frac{3a^2}{4}$
 
                                                   = $3a^2$
 
                                                   = 3 times the square of the side.