The figure shows two concentric circles and $AD$ is a chord of larger circle. prove that $AB=CD$.

A straight line is drawn cutting two equal circles and passing through the mid-point $M$ of the line joining their centres $O$ and$O^{\text{'}}$ . Prove that the chords $AB\text{ and }CD$ , which are intercepted by the two circles, are equal.

$M\text{ and }N$ are the mid-points of two equal chords $AB\text{ and }CD$ respectively of a circle with centre $O$. Prove that $\angle BMN=\angle DNM$.

$M\text{ and }N$ are the mid-points of two equal chords $AB\text{ and }CD$ respectively of a circle with centre $O$. Prove that $\angle AMN=\angle CNM$.

In the following figure; $P\text{ and }Q$ are the points of intersection of two circles with centres $O$ and $O^{\text{'}}$. If straight lines $APB\text{ and }CQD$ are parallel to $O{O}^{\text{'}}$; prove that: $O{O}^{\text{'}}=\frac{1}{2}AB$

In the following figure; $P\text{ and }Q$ are the points of intersection of two circles with centres $O$ and $O^{\text{'}}$. If straight lines $APB\text{ and }CQD$ are parallel to $O{O}^{\text{'}}$; prove that: $AB=CD$.

Two equal chords $AB\text{ and }CD$ of a circle with centre $O$, intersect each other at point $P$ inside the circle. Prove that :$AP=CP$.

Two equal chords $AB\text{ and }CD$ of a circle with centre $O$, intersect each other at point $P$ inside the circle. Prove that :$BP=DP$.

In the following figure, $OABC$ is a square. A circle is drawn with $O$ as centre which meets $OC\text{ at }P\text{ and }OA\text{ at }Q$. Prove that : $∆OPA\cong ∆OQC$.