Given that the vertices A, B, C of a quadrilateral ABCD lie on a circle. Also `angleA + angleC = 180^@` , then prove that the vertex D also lie on the same circle.

If a quadrilateral ABCD, the bisector of angleC & angleD intersect at O. Prove that angleCOD=(1)/(2)(angleA+angleB)

A quadrilateral ABCD is drawn to circumscribe a circle (see the given figure). Prove that AB+CD = AD+BC.

The vertices of a quadrilateral are A(1,0), B(7,0), C(6,3) and D(2,3). Find it's area

If on a given base B C , a triangle is described such that the sum of the tangents of the base angles is m , then prove that the locus of the opposite vertex A is a parabola.

Figure shows a container filled with a liquid of density rho . Four points A, B, C and D lie on the vertices of a vertical square. Points A and C lie on a vertical line and points B and D lies on a horizontal line. Choose the correct statement(s) about the pressure at the four points.

In the given figure, A, B and C are three points on a circle with centre O such that angle BOC = 30 ^(@) and angle AOB = 60^(@). If D is a point on the circle other than the are ABC, find angle ADC.

ABC is an equilateral triangle such that the vertices B and C lie on two parallel at a distance 6. If A lies between the parallel lines at a distance 4 from one of them then the length of a side of the equilateral triangle.

Three particles, each of the mass m are situated at the vertices of an equilateral triangle of side a . The only forces acting on the particles are their mutual gravitational forces. It is desired that each particle moves in a circle while maintaining the original mutual separation a . Find the initial velocity that should be given to each particle and also the time period of the circular motion. (F=(Gm_(1)m_(2))/(r^(2)))

If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, prove that it is a rectangle.

If circles are drawn taking two sides of a triangle as diameters, prove that the point of intersection of these circles lie on the third side.