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 the four vertices of a quadrilateral lie on the circumference of a circle , then it is called a …………. quadrilateral.

Two forces vec A B and vec A D are acting at vertex A of a quadrilateral ABCD and two forces vec C B and vec C D at C prove that their resultant is given by 4 vec E F , where E and F are the midpoints of AC and BD, respectively.

A triangle has its vertices on a rectangular hyperbola. Prove that the orthocentre of the triangle also lies on the same hyperbola.

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

The quadrilateral ABCD is circumsribed about a circle. Prove that AB+ CD= BC+DA.

The vertices A and D of square A B C D lie on the positive sides of x- and y-axis , respectively. If the vertex C is the point (12 ,17) , then the coordinates of vertex B are (a) (14 ,16) (b) (15, 3) (c) 17 ,5) (d) (17 ,12)

Answer any One question : If a quadrilateral ABCD is circumscribed about a circle with centre O, prove that AB + CD = BC + DA.

In a cyclic quadrilateral ABCD, find the value of tanA/2tan B/2 tan C/2 tan D/2 .

The vertex of angle in circle does not always lie on the circumference of the circle.

ABCD is a cyclic quadrilateral. The side BC of it is extended to E . Prove that the two bisectors of angleBADandangleDCE meet on the circumferncee of the circle .