1. If ABCD is a cyclic quadrilateral inscribed in a circle with center O, then prove that AB + CD = AD + BC.

2. If ABCD is a cyclic quadrilateral in a circle with center O, prove that AB + CD = BC + DA.

3. If ABCD is a quadrilateral inscribed in a circle with center O, prove that **AB + CD = BC + DA**.

4. A circle is drawn with center at point A of quadrilateral ABCD, which passes through points B, C, and D. Prove that \(\angle\)CBD + \(\angle\)CDB = \(\cfrac{1}{2}\angle\)BAD.

5. ABC is an equilateral triangle inscribed in a circle. If P is any point on the arc BC opposite to vertex A, then prove that \(AP = BP + CP\).

6. Two chords AB and CD of a circle with center O intersect each other at point P. Prove that \(\angle AOD + \angle BOC = 2\angle BPC\) If \(\angle AOD\) and \(\angle BOC\) are supplementary, then prove that the two chords are perpendicular to each other.

7. If ABCD is a cyclic quadrilateral with center O, prove that AB + CD = BC + DA.