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.

Let triangle ABC be an equilaterral triangle with side a. Altitude AE is Drown from A to BC.
∴ BE = EC = BC/2 = a/2
In ΔABE, by pythagoras theorem
AB² = AE² + BE²
⇒ a² = AE² + (a/2)² = AE² + (a²/4)
⇒ AE² = a² – (a²/4) = 3a²/4
4AE² = 3a²
⇒ 4 × (Altitude) = 3 × (Side)