Prove that in a quadrilateral the sum of all the sides is greater than the sum of its diagonals.

Prove that the sum of the sides of a quadrilateral is greater than twice of one of its diagonal.

Prove that the perimeter of a triangle is greater than the sum of its altitudes.

Prove that the perimeter of a triangle is greater than the sum of its altitudes.

Prove that the perimeter of a triangle is greater than the sum of its three medians.

Prove that in any parallelogram the sum of squares of the diagonals is twice the sum of the squares of two adjascent sides. Also show that the difference of the squares on two adjacent sides is equal to the rectangle contained by either diagonal and the projection of the other upon it.

Prove that in a triangle, other than an equilateral triangle, angle opposite the longest side is greater than 2/3 of a right angle.

Prove that the lines joining the middle points of the opposite sides of a quadrilateral and the join of the middle points of its diagonals meet in a point and bisect one another

Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals.

Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals.

In a convex hexagon, prove that the sum of all interior angle is equal to twice the sum of its exterior angles formed by producing the sides in the same order.