Question

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

Answer 1

 let x be the side of the equilateral triangle.
then altitude = x√3 / 2
4 times altitude = 4 × (x√3 / 2)^2
                       = 4 × (3x^2 / 4)
                       = 3x^2
                       = 3 times the square of side.

Answer 2

See diagram.

Use Pythagoras theorem in the right angle triangle ADC.
        AC² = AD² + DC²

a² = h² + (BC/2)² a² = h² + a² / 4

    h² = a² - a²/4 = 3a²/4

          4 h² = 3 a²