Step 1: Convert the given data into the figure.

                BL and CM are medians of ∆ABC in which ∠A = 90°

Step 2:  Make a note of all the right triangles in the figure.

               NOTE: Right triangles

                         \triangle ABC  

                         \triangle ABL

                         \triangle AMC

Step 3: Apply the Pythagoras theorem to all the right angle triangles in the figures.

             EXPLANATION:

                     \triangle ABC  

                       BC^2 = AB^2 + AC^2 ............................(1)

                       \triangle ABL

                     BL^2 = AB^2 + AL^2 ...........................(2)

                       \triangle AMC

                     CM^2 = AM^2 + AC^2 .....................................(3)

Step 4: Use the given hints in the question to prove the required equation.

             EXAMPLE:  From equation (2)

                         BL^2 = AB^2 + AL^2

                         BL^2 = AB^2 + (\frac{AC}{2})^2   (∵ BL is median, L is the midpoint of AC)

                         4BL^2 = 4AB^2 + AC^2 .......................(4)

                 From equation (3)

                     CM^2 = (\frac{AB}{2})^2 + AC^2    (∵ CM is median, M is the midpoint of AB)

                       4CM^2 = AB^2 + 4AC^2 ...........................(5)

Step 5: Add  equation (4) and (5)

             4CM^2 + 4BL^2 = 4AB^2 + AC^2 + AB^2 + 4AC^2

             4(CM^2 + BL^2) = 5AB^2 + 5AC^2

             4(CM^2 + BL^2) = 5(BC)^2 [Since equation (1)]

           Hence proved