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

Let the side of the equilateral triangle be a, and AE be the altitude of ΔABC

BE = EC = =

Applying Pythagoras theorem in ΔABE, we obtain

AB² = AE² + BE²

a² = AE² + ²

AE² = a² –

AE² =

4AE² = 3a²

4 × (Square of altitude) = 3 × (Square of one side)