Two chords $\mathrm{AB}$ and $\mathrm{AC}$ of a circle are equal. Prove that centre of circle lies on the bisector of $\angle \mathrm{BAC}$.

Centre of a circle is $O$ and $PO$ is the angle bisector of $\angle APD$ (See in the figure). Prove $AB=CD$.

Two equal chords $AB$ and $CD$ in a circle with centre at $O$ meet at right angle on point $E$. If $P$ and $Q$ are midpoints of chords $\mathrm{AB}$ and $\mathrm{CD}$ then prove that $OPEQ$ is a square.

In figure, $\mathrm{O}$ is centre of circle, $PQ$ is a chord. If$\angle PRQ=50°$, then find $\angle OPQ$?

In figure, find the value of $\angle PBO$, if $\angle AOB=50°$ and $\angle PQB=75°$?

If $O$ is the centre of a circle, then find the value of $x$ in the figure.