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

Given: In ΔABC,

AB = AC = BC

To prove: 3 AB² = 4 AD²

Theorem Used:

1) If corresponding side and two angles of two triangles are equal the triangles are said to have SAS congruency.

2) Pythagoras theorem:

In a right-angled triangle, the squares of the hypotenuse is equal to the sum of the squares of the other two sides.

Proof:

Let AD ⊥ BC

In ΔADC and ΔADB

AB = AC (given)

∠B = ∠C (each 60°)

∠ADB = ∠ADC (each 90°)

∴ ΔADC ≅ ΔADB

⇒ BD = DC

As ΔADB is a right triangle right angled at D.

By Pythagoras theorem,

AB² = AD² + BD²

⇒ AB² = AD² + (1/2 BC)²

⇒ AB² = AD² + (1/4 BC²)

As BC = AB

⇒ AB² = AD² + (1/4 AB²)

⇒ 3 AB² = 4 AD²

Hence proved.