Prove that the quadrilateral obtained by joining any two alternate vertices of a regular pentagon is cyclic.

Prove that the quadrilatral formed by joining the mid points of the sides of a quadrilateral is a parallelogram.

Prove that the line joining the centres of two intersecting circles is the perpendicular bisector of the line joining the points of intersection.

Prove that the line joining the centres of two intersecting circles is the perpendicular bisector of the line joining the points of intersectíon.

The coordinates of the vertices of a quadrilateral taken in order are '(2,1)', '(5,3),(8,7),(4,9)'. i) Find the coordinates of the midpoints of all four sides. ii) Prove that the quadrilateral got by joining these mid points is a parallelogram.

The coordinates of the verticesofa quadrilateral, taken in order, are (2,1), (5,3), (8,7), (4,9).Prove that the quadrilateral got by joining these midpoints is a parallelogram.

Prove that the quadrilateral with vertices '(1,1),(5,1),(7,8),(3,8)' is a parallelogram.

In the picture below, the top vertex of a triangle is joined to the midpoint of the bottom side of the triangle and then the midpoint of this line is joined to the other two vertices. Prove that the areas of all four triangles obtained thus are equal to a fourth of the area of the whole triangle.

In the picture below, the top vertex of a triangle is joined to the midpoint of the bottom side of the triangle and then the midpoint of this line is joined to the other two vertices. Prove that the areas of all four triangles obtained thus are equal to a fourth of the area of the whole triangle.

Prove that for any quadrilateral with diagonals perpendicular, the area.ls half the product of the diagonals: