 Hello and welcome to the session. Let us discuss the following question which says if the diagonals of a quadrilateral divide each other proportionally, prove that it is a trapezium. Let's see its solution. Consider the quadrilateral A, B, C, D in which the diagonals A, C and B, D intersect at O. Now, we are given in the question that the diagonals of the quadrilateral divide each other proportionally. That means we are given that A O upon O C is equal to B O upon O D and we need to prove that the quadrilateral A, B, C, D is a trapezium. For this, we will do one construction O E parallel to A, B. Now, let's see its proof. First of all, consider the triangle A, B, D. So, in triangle A, B, D, we have O E parallel to A, B. So, this implies that the E upon E, A will be equal to D O upon O B because if a line is drawn parallel to one side of a triangle, it will set the other two sides in distinct points. The other two sides are divided in the same ratio. So, let us mark this as one, but O upon O C is equal to O B upon O D. This is given to us. So, this implies that O C upon A O will be equal to O D upon O B. Let us mark it as two. Now, from one and two, we will get D E upon E A is equal to O D upon O B and O D upon O B is equal to O C upon A O. So, this means that D E upon E A will be equal to O C upon A O. Now, consider the triangle A D C. So, in triangle A D C, we have D E upon E A equal to O C upon A O. This we have already proved. This implies O E will be parallel to D C because if a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side. Thus, here O E is parallel to D C. Now, O E is parallel to A B by construction. Is parallel to D C proved above? Now, we know that if a line is parallel to the two given lines, then the two lines are parallel to each other. So, that means A B is parallel to D C. Now, A B C D is a quadrilateral in which one pair of opposite sides is parallel. So, that means A B C D is a trapezium because a trapezium is a quadrilateral in which one pair of opposite sides is parallel. Hence, proved. With this, we finish this session. Hope you must have enjoyed it. Goodbye, take care and have a nice day.