$\overline{ PA }$ and $\overline{ PB }$ are tangents to the $\odot( O , r)$ drawn from a point $P$ outside the circle,then ..... .

Write 'True' or 'False' and give reasons for your answer.
In the figure,$BOA$ is a diameter of a circle and the tangent at a point $P$ meets $BA$ extended at $T$. If $\angle PBO = 30^{\circ}$,then $\angle PTA$ is equal to $30^{\circ}$.

$A$ chord $PQ$ of a circle is parallel to the tangent drawn at a point $R$ of the circle. Prove that $R$ bisects the arc $PRQ$.

Write 'True' or 'False' and give reasons for your answer.
The angle between two tangents to a circle may be $0^{\circ}$.

Prove that the tangent drawn at the mid-point of an arc of a circle is parallel to the chord joining the end points of the arc.

Let $s$ denote the semi-perimeter of a triangle $ABC$ in which $BC = a, CA = b, AB = c$. If a circle touches the sides $BC, CA, AB$ at $D, E, F$ respectively,prove that $BD = s - b$.