1. ABCD is a cyclic quadrilateral. DE is a chord which acts as the external bisector of ∠BDC. Prove that AE (or extended AE) is the external bisector of ∠BAC.

2. ABCD is a cyclic quadrilateral. The bisectors of \(\angle\)DAB and \(\angle\)BCD intersect the circle at points X and Y. Prove that XY is the diameter of the circle.

3. ABCD is a cyclic quadrilateral. The angle bisectors of \(\angle\)DAB and \(\angle\)BCD intersect the circle at points X and Y respectively. If O is the center of the circle, find the value of \(\angle\)XOY.

4. Prove that the quadrilateral formed by the intersection of the angle bisectors of the four angles of any quadrilateral is a cyclic quadrilateral.

5. ABCD is a cyclic quadrilateral. Side AB is extended to point X. If \(\angle XBC = 82^\circ\) and \(\angle ADB = 47^\circ\), find the value of \(\angle BAC\).

6. From an external point \(A\), two tangents are drawn to a circle centered at \(O\), touching the circle at points \(B\) and \(C\) respectively. Prove that the line \(AO\) is the perpendicular bisector of the line segment \(BC\).

7. ABCD is a cyclic quadrilateral. The sides AD and AB are extended to E and F, respectively. If \(\angle\) CBF = 120°, then find the measure of \(\angle\) CDE.

8. O is the center of the circle, and AB is its diameter. ABCD is a cyclic quadrilateral. Given that \(\angle\)ABC = 65° and \(\angle\)DAC = 60°, find the measure of \(\angle\)BCD.

9. In the given figure, QS is the bisector of \(\angle\)PQR. Given that \(\angle\)SQR = 35° and \(\angle\)PRQ = 32°, determine the measure of \(\angle\)QSR.

10. AB and AC are two equal chords of a circle. Prove that the bisector of \(\angle\)BAC passes through the center of the circle.

11. In a circle, triangle ABC is inscribed. AX, BY, and CZ are the angle bisectors of \(\angle BAC\), \(\angle ABC\), and \(\angle ACB\) respectively, and they meet the circle at points X, Y, and Z respectively. Prove that AX is perpendicular to YZ.

12. In a circle, triangle ABC is inscribed. The angle bisectors of \(\angle BAC\), \(\angle ABC\), and \(\angle ACB\) meet the circle at points X, Y, and Z respectively. Prove that in triangle ΔXYZ, \(\angle YXZ = 90^\circ - \frac{\angle BAC}{2}\).

13. In the adjacent figure, BX and CY are the bisectors of \(\angle\)ABC and \(\angle\)ACB respectively. Given that AB = AC and BY = 4 cm, find the length of AX.

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

15. ABCD is a cyclic quadrilateral. The angle bisectors of ∠DAB and ∠BCD intersect the circle at points X and Y, respectively. Prove that XY is a diameter of the circle.

16. Let us prove that the quadrilateral formed by the intersection of the internal angle bisectors of the four angles of a quadrilateral is a cyclic quadrilateral.

17. From an external point \(A\), two tangents are drawn to a circle centered at \(O\), touching the circle at points \(B\) and \(C\) respectively. Prove that the line \(AO\) is the perpendicular bisector of the chord \(BC\).

18. I have drawn a cyclic quadrilateral ABCD, and extended its side BC up to point E. Prove that the bisectors of ∠BAD and ∠DCE meet on the circle.

19. From an external point \(P\), two tangents \(PS\) and \(PT\) are drawn to a circle with center \(O\). \(QS\) is a chord of the circle that is parallel to \(PT\). If \(\angle SPT = 80^\circ\), then what is the measure of \(\angle QST\)?