ABC is a cyclic triangle and the bisectors of its angles intersect the circumference of the circle at X,Y and Z. prove that AX is perpendicular to YZ.

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.

The internal and external bisectors of the vertical angle of a triangle intersect the circumcircle of the triangle at the points P and Q. Prove that PQ is a diameter of the circle.

The triangle ABCis inscribed in a circle, AX, BY and CZ of the angles angle BAC, angle ABC and angle ACB , intersect at the point X,Y,Z on the circle respectively. Prove that AX is perpendicular to XZ.

Prove that the internal bisector of any angle of a cyclic quadrilateral and the external bisector of its opposite angle intersect each other on the circumference of the circle.

If the angle - bisector of two intersecting chords of a circle passes through its centre , then prove that the chords are equal .

ABC is an acute-angled triangle .AP is the diameter of the circumcircle of the triangle ABC, EB and CF are perpendiculars on AC and AB respectiely and they intersect each other at the point other at the point Q. Prove that BPCQ is a parallelogram.

I is the centre of the incircle of Delta ABC, produced AI intersects the circumercile of that triangle at the point P. Prove that PB=PC=PI

Prove analytically that the bisectors of the interior angles of a triangle are concurrent.

Bisectors of angles A, B and C of a triangle ABC intersect its circumcircle at D, E and F respectively. Prove that the angles of the triangle DEF are 90^@-1/2A , 90^@-1/2B and 90^@-1/2C