$A$ line that intersects a circle at two distinct points is called a $\ldots \ldots \ldots \ldots$ of the circle.

Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.

In the following figure,if $AB = 10$,then $AC = \ldots$

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.

Two concentric circles having radii $41$ and $40$ are given. The chord of the circle with larger radius touches the circle with smaller radius. Find the length of the chord.

$A$ triangle $ABC$ is drawn to circumscribe a circle of radius $4 \, cm$ such that the segments $BD$ and $DC$ into which $BC$ is divided by the point of contact $D$ are of lengths $8 \, cm$ and $6 \, cm$ respectively (see figure). Find the sides $AB$ and $AC$ (in $cm$).