A triangle has an area of 2. The lengths of its medians equal the lengths of the sides of a second triangle. The lengths of the medians of the second triangle equal the lengths of the sides of a third triangle. In general, the medians of the zth triangle have the same lengths as the sides of the (z+1) triangle. Find the limit of the sum of the areas of all the triangles thus formed in this infinite sequence.
I tried to solve it by creating a right triangle with lengths 2 and 2 but finding the medians became very difficult so I discarded that method. Is there a cleaner method for solving this?