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

In a parallelogram ABCD, the bisectors of the consecutive angles angleA and angleB intersect at P. Show that angleAPB=90^(@) .

ABCD is a cyclic quadrilateral .The side BC is extended to E .The bisectors of the angles angleBADandangleDCE intersect at the point P .Prove that angleADC=angleAPC .

In Delta ABC, angle B le angle C. If the interior bisector of angle BAC be AP and AQbotBC. then prove that angle PAQ=(1)/(2)(angle B-angleC)

Draw a circle with the point a of quadrilateral ABCD as centre which passes through the points B,C and D.Prove that angle CBD+ angleCDB=(1)/(2) angle BAD .

ABCD is a cyclic quadrilateral. The bisectors of angleDAB and angleBCD intersect the circle at the points X and Y respectively. Prove that XY is a diameter of the circle.

In a isosceles triangle ABC , angleB is right angle. Bisector of angleBAC intersect BC at D. Prove that CD^2= 2BD^2 .

The bisector of the interior angles angleB and angle C of the DeltaABC , meet at H. Prove that angleBHC=90^@+(1)/(2)angleBAC .

Two oppsite angles of the quadrilateral ABCD are supplementary and aC bisects angle BAD . Prove that BC Cand CD are equal.

If the bisector of an angle of a triangle also bisects the opposite side, prove that the triangle is isosceles.

ABCD is a cyclic quarilateral. The bisectors of angleaandangleC in tersect the circle at the points E and F respectively . Prove that EF is a diameter of the circle.