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

Step 1: According to the given data built an imaginary diagram
ABC be equilateral triangle.
AD be perpendicular bisector from A on to BC.
So BD = DC = \frac{1}{2} BC
Step 2: Apply the Pythagoras theorem to an right angle triangle in equilateral.
EXAMPLE: Consider ADC is a right angle triangle.
So, AC^2 = AD^2 + DC^2
AC^2 = AD^2 + (\frac{1}{2} BC)^2
It is an equilateral triangle so AB = BC = AC
AC^2 = AD^2 + (\frac{1}{2} AC)^2
AD^2 = AC^2 - \frac{(AC)^2}{4}
4AD^2 = 4 AC^2 - AC^2
4 AD^2 = 3 AC^2
Hence proved