the segments joining the midpoints of opposite sides of any quadrilateral bisect each other. kindly prove this

Answer 1

Do a simple construction. Draw the diagonals of the quadrilateral.
Now A line joining the midpoints of two adjacent sides will be parallel to a diagonal by inverse of basic proportionality theorem. By a similar argument we get two pairs of segements parallel to the corresponding diagonals. Thus the figure obtained by joining the midpoints is a parallelogram. Diagonals of this parallelogram are nothing but the lines joining opposite sides' midpoints. Since diagonals of a parallelogram bisect each other, the two line segments will bisect each other.