Let S be a square of unit area. Consider any quadrilateral which has one vertex on each side of S. If `a , b , c and d` denote the lengths of sides of the quadrilateral, prove that `2 ≤ a_2 + b_2 + c_2 + d_2 le 4`

Let alt=blt=c be the lengths of the sides of a triangle. If a^2+b^2 < c^2, then prove that triangle is obtuse angled .

a triangle A B C with fixed base B C , the vertex A moves such that cosB+cosC=4sin^2(A/2)dot If a ,b and c , denote the length of the sides of the triangle opposite to the angles A , B and C , respectively, then (a) b+c=4a (b) b+c=2a (c) the locus of point A is an ellipse (d) the locus of point A is a pair of straight lines

The area of any cyclic quadrilateral ABCD is given by A^(2) = (s -a) (s-b) (s-c) (s-d) , where 2s = a + b ++ c + d, a, b, c and d are the sides of the quadrilateral Now consider a cyclic quadrilateral ABCD of area 1 sq. unit and answer the following question The minium perimeter of the quadrilateral is

The area of any cyclic quadrilateral ABCD is given by A^(2) = (s -a) (s-b) (s-c) (s-d) , where 2s = a + b ++ c + d, a, b, c and d are the sides of the quadrilateral Now consider a cyclic quadrilateral ABCD of area 1 sq. unit and answer the following question When the perimeter is minimum, the quadrilateral is necessarily

The area of any cyclic quadrilateral ABCD is given by A^(2) = (s -a) (s-b) (s-c) (s-d) , where 2s = a + b ++ c + d, a, b, c and d are the sides of the quadrilateral Now consider a cyclic quadrilateral ABCD of area 1 sq. unit and answer the following question The minimum value of the sum of the lenghts of diagonals is

If a,b,c,d,be the sides of a quadrilateral and g(x)=f[f{f(x)}], where f(x)=1/(1-x), d^2/(a^2+b^2+c^2) is equal to

If A,B,C,D are the successive angles of a cyclic quadrilateral, then- (i) cos((A+C)/2) + cos((B + D)/2) =

If a, b, c, d are in continued proportion, then prove that (b -c)^(2) + (c -a)^(2) + (b-d)^(2) = (a -d)^(2)

If a,b and c are constant of motion and s = at^(2)+bt+ c , then prove that 4a(s-c) = v^(2) - b^(2) .

If the length of a focal chord of the parabola y^2=4a x at a distance b from the vertex is c , then prove that b^2c=4a^3 .