Prove that the quadrilateral formed by joining the mid-points of the sides of a rectangle is a rhombus, but not a square.

Prove that the quadrilateral formed by joining the mid-points of the sides of a square is also a square.

Prove that the quadrilateral formed by joining the mid-points successively of any quadrilateral is also a parallelogram. Also, prove that the perimeter of that parallelogram is equal to the sum of that quadrilateral's diagonals.

Prove that the area of the parallelogram produced by joining the mid-points of the sides of a quadrilateral taken in order is half the area of the quadrilateral.

Show that the figure formed by joining the midpoints of sides of a rhombus successively is a rectangle.

Using the properties of vectors, prove that the straight line, obtained by joining the mid-poings of two sides of a triangle , is half of the length of third side.

Prove that the quadrilateral obtained by successively joining the mid -points of a parallelogram is also a parallelogram.

Prove that the line segment joining the mid-points of the lateral sides of any trapezium is parallel to any of its parallel sides.

Prove analytically that the area of a triangle is four times that of the triangle obtained by joining the mid - points of the sides of the given triangle .

Show that the straight line joining the mid - points of two sides of a triangle is equal to half the third side .

Find the area of the triangle formed by joining the mid-points of the sides of the triangle whose vertices are (0, -1), (2, 1) and (0, 3). Find the ratio of this area to the area of the given triangle.