In a circle with centre $P$,$AB$ is a diameter and point $C$ is a point other than $A$ and $B$. If the radius of the circle is $34 \, cm$ and $AC = 32 \, cm$,then $BC = $ .......... $cm$.

Write True or False and justify your answer in each of the following:
Two chords $AB$ and $AC$ of a circle with centre $O$ are on the opposite sides of $OA$. Then $\angle OAB = \angle OAC$.

Two chords $AB$ and $AC$ of a circle subtend angles equal to $90^{\circ}$ and $150^{\circ}$,respectively,at the centre. Find $\angle BAC$,if $AB$ and $AC$ lie on the opposite sides of the centre.

In the same plane,two circles can have a maximum of ............ common chords.

Bisectors of angles $A, B$ and $C$ of a triangle $ABC$ intersect its circumcircle at $D, E$ and $F$ respectively. Prove that the angles of the triangle $DEF$ are $90^{\circ}-\frac{1}{2} A, 90^{\circ}-\frac{1}{2} B$ and $90^{\circ}-\frac{1}{2} C$.

The circumcentre of the triangle $ABC$ is $O$. Prove that $\angle OBC + \angle BAC = 90^{\circ}$.