Prove that a parallelogram which is not a rectangle is not cyclic.

Prove that in any parallelogram, the 'sum of the squares of all sides is equal to the sum of the squares of the diagonals'.

ABCD is a parallelogram with A as the origin. vecb and vecd are position vector of B and D respectively. If absvec(AC)=absvec(BD) , show that ABCD is a Rectangle.

The figure shows a parallelogram with the coordinates of its vertices: Prove that x_1+x_3=x_2+x_4andy_1+y_3=y_2+y_4

Prove that any non-isosceles trapezium is not cyclic.

Consider the statement "there exists a rectangle which is not a square " Its negation is ?

Prove that of all rectangles with given area square has minimum perimeter?

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

In the picture below, two vertices of a parallelogram are joined to the mid points of two sides. Prove that these lines divides the diagonal in the picture into three equal parts. .