X OM - I - Task 8

The sides of a triangle are the medians of another triangle. Calculate the ratio of the areas of both triangles. Can a triangle be constructed from the medians of any triangle?
Let $ S $ be the centroid of triangle $ ABC $. Extend the median $ AD $ by a segment $ DM = SD $ (Fig. 15). We obtain triangle $ BSM $, in which $ BS = \frac{2}{3} BE $, $ SM = \frac{2}{3} AD $, and $ BM = SC = \frac{2}{3} CF $ (since quadrilateral $ SBMC $ is a parallelogram). Triangle $ BSM $ is thus similar to triangle $ T $ formed by the medians of triangle $ ABC $, with a scale factor of $ \frac{2}{3} $; the area of triangle $ BSM $ is therefore $ \frac{4}{9} $ of the area of $ T $. However, the area of $ \triangle BSM = \text{area of }\triangle ASB = \frac{1}{3} \cdot \text{area of }\triangle ABC $. It follows that $ \text{area of }T = \frac{3}{4} \cdot \text{area of }\triangle ABC $.