Write True or False and justify your answer in each of the following: Through three collinear points a circle can be drawn.

In the given figure,$ABCD$ is a cyclic quadrilateral in which diagonals $AC$ and $BD$ intersect at $M.$ If $\angle BAC = 50^{\circ}$ and $\angle ADB = 45^{\circ}$,then find $\angle ABC$. (in $^{\circ}$)

In the given figure,if $\angle ABC = 45^{\circ}$,prove that $OA \perp OC$.

Prove that equal chords of congruent circles subtend equal angles at their centres.

$AB$ and $CD$ are two parallel chords of a circle with centre $P$. Centre $P$ is not between the two chords $AB$ and $CD$. If $AB = 40\,cm$,$CD = 30\,cm$ and the radius of the circle is $25\,cm$,find the distance between $AB$ and $CD$. (in $,cm$)

$AB$ and $AC$ are two equal chords of a circle. Prove that the bisector of the angle $BAC$ passes through the centre of the circle.