 Fine, how are you all today? The question to be discussed today is D, E and F are respectively the midpoints of side B, C, C and AB of triangle ABC. Show that first of all, B, D, E, F is a parallelogram. Now this is the figure which we need to refer and the B part which we need to prove is that the area of triangle D, E, F is one-fourth the area of triangle ABC and the C part is that the area of B, D, E, F is half the area of this full ABC triangle. So let us proceed on with our proof. Now we know that in triangle ABC by the midpoint theorem that says that the line joining two sides of a triangle is parallel to the third side and half of it. So we can say that E, F is parallel to BC that is by midpoint theorem, right? And since we can see that D is the midpoint of BC so E, F will be parallel to BD also so we can say that therefore E, F is parallel to BD. Also in the same triangle we can see that E, D is parallel to AB again by midpoint. So we can say that therefore E, D is parallel to half of this side that is A, F that is parallel to A, F also E, D is parallel to E, B also. So we can say that from one and two we have now we can see that E, F is parallel to BD and E, D is parallel to E, B, F, B. So from one and two we can say that BD, E, F parallelogram, right? This completes the first part of the proof. The second part requires us to answer that proof that area of this small triangle that is DEF is one fourth the area of ABC. Now we can say that similarly as we prove that BDEF is a parallelogram we can say that F, T, C, E and AF, DE are parallelograms, right? So we know that diagonal of a parallelogram divides it into angles of equal area. Since F, T is a diagonal to the parallelogram BDEF that means it is dividing it into two triangles of equal area, right? So we can say that since we know all these three are the parallelogram we can say that area of DE is equal to area of DE, that is F, BD is equal to DEF again, area of DEC, DEC is again equal to area of DEF, area of AFE is equal to area of DEF. Now here the diagonal of AFDE, this triangle, this is the diagonal so again this triangle will be equal to the area of this triangle. This from all these three statements that I have written over here says that all these three triangles are equal to the same triangle so we can say that therefore area of FB is equal to area of DEC is equal to area of AFE is equal to area of DEF. Now if you carefully observe this triangle ABC is made up of all these four triangles itself so this implies that area of DEF is one fourth the area of ABC, right? This completes our second part's proof and proceeding on with the last and final part we know that area of parallelogram BDEF is what? It is twice this triangle or this triangle so we can say that it's twice the area of DEF that is now area of DEF as we have proved in first part can be written as one fourth area of ABC so it is further equal to half the area of ABC, right? This completes all the three proofs that we were required to do hope you enjoyed and have a good day.