If $\square ABCD$ is a cyclic quadrilateral and also a rectangle,and if $AB = 5$ and $BC = 12$,then $AC = \ldots$

Write 'True' or 'False' and give reasons for your answer.
$AB$ is a diameter of a circle and $AC$ is its chord such that $\angle BAC = 30^{\circ}$. If the tangent at $C$ intersects $AB$ extended at $D$,then $BC = BD$.

If a hexagon $ABCDEF$ circumscribes a circle,prove that $AB + CD + EF = BC + DE + FA$.

$\overline{PA}$ is a tangent to $\odot(O, 8)$ drawn from a point $P$ outside the circle. If $m\angle AOP = 45^\circ$,then $AP = \ldots$

Two tangents $PQ$ and $PR$ are drawn from an external point $P$ to a circle with centre $O$. Prove that $QORP$ is a cyclic quadrilateral.

If a circle touches the side $BC$ of a triangle $ABC$ at $P$ and extended sides $AB$ and $AC$ at $Q$ and $R$ respectively,prove that $AQ = \frac{1}{2}(BC + CA + AB)$.