$\overline{ PA }$ and $\overline{ PB }$ are the tangents to $\odot( O , r)$ drawn from a point $P$ outside a circle. If $m \angle APB = 65^{\circ}$,then $m \angle AOB = \ldots \ldots \ldots . .$ (in $^{\circ}$)

If $a, b, c$ are the sides of a right triangle where $c$ is the hypotenuse,prove that the radius $r$ of the incircle of the triangle is given by $r = \frac{a + b - c}{2}$.

Point $P$ lies in the exterior of $\odot(O, r)$. $\overleftrightarrow{PQ}$ touches the circle at $Q$. If $PO = 26$ and $PQ = 10$,then find the diameter of the circle.

From an external point $P$,two tangents,$PA$ and $PB$ are drawn to a circle with centre $O$. At one point $E$ on the circle,a tangent is drawn which intersects $PA$ and $PB$ at $C$ and $D$,respectively. If $PA = 10 \, cm$,find the perimeter of the triangle $PCD$ (in $cm$).

In the following figure,find $m \angle PBA$.

In the figure,common tangents $AB$ and $CD$ to two circles intersect at $E$. Prove that $AB = CD$.