Two equal chords $\mathrm{AB}$ and $\mathrm{CD}$ are drawn in a circle with centre $O$. Show that $∆\mathrm{OAB}$ and $∆\mathrm{OCD}$ are congruent.

$\mathrm{ABCD}$ is a square. $A$ is joined to $X$ on $BC$ and $D$ to a point $Y$ on $AB$ as shown in the figure. If $AX=BY$, prove that $AX$ and $DY$ are at right angles.