How to prove that the centre of a circle is the same as the intersection point of the diagonals of its cyclic quadrilateral ?

Answer 1

Not even true.

Counterexample:

draw any circle
Pick point P, Q, and R on the right side of the circle, near each other.
Pick S opposite those on the left side.
Join PQRS to form a cyclic quadrilateral.
Draw the diagonals.
Does the intersection of these two diagonals come anywhere near the centre of your circle?

The statement would only be true if your quad was a rectangle.
Then the centre would be the point of intersection of the diagonals.
Now "drag" one of the vertices of the rectangle along the circle.
One of the diagonals would stay the same, but the other one would change.
So their intersection would no longer be the same, disproving your above statement.