In quadrilateral $ABCD$,if $\angle A : \angle B : \angle C : \angle D = 3 : 2 : 4 : 5$,then identify the type of quadrilateral $ABCD$.

$A, B$ and $C$ are three points on a circle. Prove that the perpendicular bisectors of $AB, BC$ and $CA$ are concurrent.

In a circle with centre $P$,$AB$ is a diameter and $ABCD$ is a cyclic quadrilateral. If $\angle ADC = 150^{\circ}$,then find $\angle BAC$. (in $^{\circ}$)

Three girls Reshma,Salma and Mandip are playing a game by standing on a circle of radius $5 \, m$ drawn in a park. Reshma throws a ball to Salma,Salma to Mandip,Mandip to Reshma. If the distance between Reshma and Salma and between Salma and Mandip is $6 \, m$ each,what is the distance between Reshma and Mandip?

In the same plane,two circles can have a maximum of ............ common chords.

$ABCD$ is a cyclic quadrilateral. If $AB \parallel CD$,then prove that $BC = AD$.