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.

Given: ΔABC is an Equilateral Triangle. AB=BC=AC CD Perpendicular to AB
To Prove: 3AC²=4CD²
Proof: In ΔADC
By Pythagoras Theorem, AC²=AD²+DC² (1)
In ΔBDC
By Pythagoras Theorem, BC²=DC²+BD² (2)
Adding 1) and 2) We get, 2AC²=2DC²+AD²+BD² (Since AC=BC)
Since, AD=BD=1/2AC (Since AC=AB)
∴2AC²=2DC²+1/2AC²
∴2AC²−1/2AC²=2DC²
∴3/2AC²=2DC²
∴3AC²=4CD²