Prove that the area of a regular polygon hawing `2n` sides, inscribed in a circle, is the geometric mean of the areas of the inscribed and circumscribed polygons of `n` sides.

The ratio of the area of a regular polygon of n sides inscribed in a circle to that of the polygon of same number of sides circumscribing the same is 3:4. Then the value of n is

the sum of the radii of inscribed and circumscribed circules for an n sides regular polygon of side a, is

Prove that the area of the equilateral triangle described on the side of a square is half the area of the equilateral triangles described on its diagonal.

Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.

I_(n) is the area of n sided refular polygon inscribed in a circle unit radius and O_(n) be the area of the polygon circumscribing the given circle, prove that I_(n)=O_(n)/2(1+sqrt(1-((2I_(n))/n)^(2)))

A circle is inscribed in an equilateral triangle of side adot The area of any square inscribed in this circle is ______.

Regular pentagons are inscribed in two circles of radius 5 and 2 units respectively. The ratio of their areas is

Prove that the sum of the radii of the radii of the circles, which are, respectively, inscribed and circumscribed about a polygon of n sides, whose side length is a , is 1/2acotpi/(2n)dot

Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.

Find the number of diagonals in the convex polygon of n sides .