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