In the figure,if $AOB$ is a diameter of the circle and $AC = BC$,then $\angle CAB$ is equal to (in $^{\circ}$)

Prove that the line of centres of two intersecting circles subtends equal angles at the two points of intersection.

Two circles intersect at two points $B$ and $C$. Through $B$,two line segments $ABD$ and $PBQ$ are drawn to intersect the circles at $A, D$ and $P, Q$ respectively (see figure). Prove that $\angle ACP = \angle QCD$.

$AC$ and $BD$ are chords of a circle which bisect each other. Prove that $AC$ and $BD$ are diameters.

Two circles intersect at two points $A$ and $B$. $AD$ and $AC$ are diameters to the two circles (see Fig.). Prove that $B$ lies on the line segment $DC$.

Let the vertex of an angle $\angle ABC$ be located outside a circle and let the sides of the angle intersect equal chords $AD$ and $CE$ with the circle. Prove that $\angle ABC$ is equal to half the difference of the angles subtended by the chords $DE$ and $AC$ at the centre.