XVI OM - III - Task 5

Points $ A_1 $, $ B_1 $, $ C_1 $ divide the sides $ BC $, $ CA $, $ AB $ of triangle $ ABC $ in the ratios $ k_1 $, $ k_2 $, $ k_3 $. Calculate the ratio of the areas of triangles $ A_1B_1C_1 $ and $ ABC $.
According to the assumption






Hence






Let the symbol $ p (ABC) $ denote the area of triangle $ ABC $ (Fig. 21). Since






then






By applying the theorem that the ratio of the areas of two triangles having a common angle equals the ratio of the products of the sides of each triangle forming that angle, we obtain






Substituting into equation (2) the values of the ratios (1) gives the equality






after transforming the right side, we obtain