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