 Hi and welcome to the session. Let us discuss the following question. Question says, in figure 6.19, DE is parallel to AC and DF is parallel to AE. Prove that BF upon FE is equal to BE upon EC. This is figure 6.19. First of all, let us understand basic proportionality theorem. It states that if a line is drawn parallel to one side of a triangle, intersecting the other two sides at two distinct points, then it divides the other two sides in the same ratio. If we are given a triangle ABC in which DB is parallel to BC and DE intersects AB at D and AC at A. So, it is intersecting AB and AC at two distinct points. So, basic proportionality theorem states that AD upon DB is equal to AE upon BC. So, we can write DE is parallel to BC, then AD upon DB is equal to AE upon BC. This is our key idea to solve the given question. Let us start the solution. We are given in triangle BC AE, DE is parallel to AC. So, by basic proportionality theorem, BD upon DA is equal to BE upon EC. Now we can write in triangle BC AE, DE is parallel to AC. Now by basic proportionality theorem, we get BD upon DA is equal to BE upon EC. Now let us consider the triangle BEA. In triangle BEA, FT is parallel to AE. Also, FT is intersecting BE and BA at two distinct points. So, this implies by basic proportionality theorem, BD upon DA is equal to BE upon FE. So, we can write in triangle BEA, DF is parallel to AE. This is given in the question. Now, by basic proportionality theorem, we get BD upon DA is equal to BE upon FE. Now clearly we can see in triangle BCA, BD upon DA is equal to BE upon EC. Let us name this expression as 1 and in triangle BEA, BD upon DA is equal to BE upon FE. Let us name this expression as 2. Now we can write from expression 1 and 2, we get BE upon EC is equal to BE upon FE. Since both the ratios are equal to BE upon DA, so we can write BE upon EC is equal to BE upon FE. So, now we have proved BE upon FE is equal to BE upon EC. This is our required answer and it is proved. This completes the session. Hope you understood the session. Have a nice day.