1. ABCD is a cyclic quadrilateral. The extended lines AB and CD intersect at point P. Prove that PA × PB = PC × PD.

2. ABCD is a cyclic quadrilateral. The extended lines AB and DC intersect at point P. Prove that: \[ PA \cdot PB = PC \cdot PD \]

3. ABCD is a cyclic quadrilateral. The extended sides AB and DC intersect at point P. Prove that: \[ PA \cdot PB = PC \cdot PD \]

4. ABCD is a cyclic quadrilateral. If the extended sides AB and DC intersect at point P, prove that PA.PB = PC.PD.

5. ABCD is a cyclic quadrilateral. The extended sides \( AB \) and \( DC \) intersect at point \( P \). Prove:\( PA \cdot PB = PC \cdot PD \).

6. ABCD is a cyclic quadrilateral. The extended sides AB and DC intersect each other at point P. Prove that \[ PA \cdot PB = PC \cdot PD. \]

7. ABCD is a cyclic quadrilateral. If the sides AB and DC are extended, they meet at point P; and if the sides AD and BC are extended, they meet at point R. The circumcircles of triangles BCP and CDR intersect at point T. Prove that the points P, T, and R lie on a straight line.

8. ABCD is a cyclic quadrilateral. The sides AB and DC are extended to meet at point P, and the sides AD and BC are extended to meet at point Q. If ∠ADC = 85° and ∠BPC = 40°, then find the measures of ∠BAD and ∠CQD.

9. ABCD is a cyclic quadrilateral. The side BA is extended up to point F. AE is drawn parallel to CD, and ∠ABC = 92°, ∠FAE = 20°. Find the measure of ∠BCD.

10. 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 \(\angle BAC\).

11. 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.

12. If the cyclic quadrilateral ABCD is extended along sides AB and DC, they intersect at point P. Prove that \( PA \cdot PB = PC \cdot PD \).

13. 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.

14. 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.

15. On a circle with center O, A is a point on the circle. A tangent is drawn at point A, and point X is any point on that tangent. A secant drawn from point X intersects the circle at points Y and Z. If P is the midpoint of segment YZ, prove that XAPO (or XAOP) is a cyclic quadrilateral.

16. ABCD is a cyclic quadrilateral. The side AB is extended up to point X, and it is measured that ∠XBC = 82° and ∠ADB = 47°; calculate and write the value of ∠BAC.

17. In the adjacent figure, ABCD is a cyclic quadrilateral. BA is extended to point F. AE ∥ CD, ∠ABC = 92°, and ∠FAE = 20°. Find the measure of ∠BCD.

(a) 20° (b) 88° (c) 108° (d) 72°