Exercise

 If $ABC$ is an equilateral triangle inscribed in a circle and $P$ be any point on a minor $\mathrm{arc} BC$ which does not coincide with $B$ or $C$, prove that $PA$ is angle bisector of $\angle \text{BPC}$.

In the given fig., $AB$ and $CD$ are two chords of a circle intersecting each other at point $E$. Prove that $\angle AEC=\frac{1}{2}$ (Angles subtended by an $\text{arc CXA}$ at the centre $+$ angle subtended by arc $DYB$ at the centre.)

If bisectors of opposite angles of a cyclic quadrilateral $ABCD$ intersect the circle, circumscribing it at the points $P$ and $Q$, prove that $PQ$ is a diameter of the circle.

A circle has radius $\sqrt{2 }\mathrm{cm}.$ It is divided into two segments by a chord of length $2 \mathrm{cm}.$

Prove that the angle subtended by the chord at a point in major segment is ${45}^o$.

 Two equal chords$AB$ and $CD$ of a circle when produced intersect at a point $P$. Prove that $PB=PD$.

$AB$ and $AC$ are two chords of a circle of radius $r$ such that $AB=2AC$. If $p$ and $q$ are the distances of $AB$ and $AC$ from the centre, prove that $4q^2=p^2+3r^2$.

In Fig. $10.20$, $O$ is the centre of the circle, $\angle BCO={30}^o$. Find $x$ and $y$.

$O$is the centre of the circle, $BD=OD$ and $CD\perp AB$. If $\angle CAB=k°$, then find the value of $k.$