Question

Prove that in an equilateral triangle, three times the square of a side is equal to four times the square of its altitudes.

Answer 1

Let ABC be an equilateral triangle and let `AD ⊥  BC`.

In  Δ ADB and Δ ADC we have

`AB=AC`

`∠ B =∠ C`

And  ` ∠ ADB = ∠ADC`

So , `BD =DC`

`⇒ BD =DC = 1/2 BC`

Since Δ ADB  is a right triangle right-angled at D. So

`AB^2=AD^2+BD^2`

`AB^2=AD^2+(1/2BC)^2`

`AB^2=AD^2+(AB^2)/4`

`3/4 AB^2 =AD^2`

`3AB^2=4AD^2`

Hence proved.