ABCD is a cyclic quadrilaterla. BC is produced to E. The bisectors of `angle ABD and angle DCE ` intersect at the points P. Prove that `angle ADC= angle APC`

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 .

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.

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.

If angle A and angle P are acute angles such that sin A =sin P then prove that angle A =angle P

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

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 .

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

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

In Delta ABC,AB= AC and E is any point on the extended BC. The circumeircel of Delta ABC intersects AE at the point D. Prove that angle ACD= angle AEC .

In DeltaABC , the point at which the bisectors of the angles /_ABC and /_BAC intersect is called the incentre of the triangle.