Bidyut Guha Thakurta and Amitava Mitra Solutions for Exercise 1: EXERCISE 3

In the cyclic quadrilateral $ABCD$ the side $BC$ is produced to $E$. The centre of the circle is $O$ and if $\angle DCE=80°$ find $\angle BAD$ and $\angle BOD$.

$RS$ is the diameter of a circle. The chord $MN$ of the circle is parallel to $RS$. If $MRN=40°$ what is the value of $RNM ?$

$AP$ and $AQ$ are two chord of a centre $O$. The mid point of $AP$ and $AQ$ are $R$ and $S$. Prove that $O, R, A$ and $S$ are concyclic.

In a cyclic quadrilateral $ABCD$, prove that the internal bisector of $\angle A$ and the external bisector of $\angle C$ meet at a point on the circumference.

$ABC$ is an isosceles triangle of which $AB=AC$. If the circle through $B$ and $C$ meets $AB$ at $E$ and $AC$ at $F$, prove that $BCEF$ is an isosceles trapezium.

$ABCD$ is  a cyclic quadrilateral of which $AB\parallel DC$ and the side $AB$ is diameter of the circle. If $\angle BAC=25°$ then $\angle DAC$ is

$ABCD$ is a cyclic quadrilateral. If the ratio of measure of the angle $A, B, C$ be $1 : 2 : 3$, the the ratio of the measure of angles $B$ and $D$ is _____

$PQRS$ is a cyclic quadrilateral. $O$ is the centre of the circle and $QR$ is produced to $T$. If measure of $\angle QOS=120°$, then measure of $\angle SRT=60°$.