The perpendicular from the centre of a circle to a chord ........... the chord.

In a cyclic quadrilateral $ABCD$,$\angle A = 4 \angle C$ and $\angle B = 3 \angle D$. Find all the angles of $ABCD$.

If $P, Q$ and $R$ are the mid-points of the sides $BC, CA$ and $AB$ of a triangle $ABC$ and $AD$ is the perpendicular from $A$ on $BC$,prove that $P, Q, R$ and $D$ are concyclic.

$AD$ is a diameter of a circle and $AB$ is a chord. If $AD = 34 \, cm$ and $AB = 30 \, cm$,the distance of $AB$ from the centre of the circle is (in $cm$):

Write True or False and justify your answer in each of the following:
$A$ circle of radius $3 \, cm$ can be drawn through two points $A$ and $B$ such that $AB = 6 \, cm$.

If non-parallel sides of a trapezium are equal,prove that it is cyclic.