$\mathrm{AB}$ and $\mathrm{AC}$ are two equal chords of a circle with centre $\mathrm{O}$. If $\mathrm{OD}\perp \mathrm{AB}$ and $\mathrm{OE}\perp \mathrm{AC}$, then prove that $∆\mathrm{ADE}$ is isosceles triangle.

Chords which are equidistant from the centre of a circle are equal. Prove.

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.