Show that the diagonals of a square are equal and perpendicular to each other.

$PQ$ and $RS$ are two equal and parallel line segments. Any point $M$ not lying on $PQ$ or $RS$ is joined to $Q$ and $S$ and lines through parallel to $QM$ and through $R$ parallel to $SM$ meet at $N$. Prove that line segments $MN$ and $PQ$ are equal and parallel to each other.

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In the given figure, $ABCD$ is a parallelogram and $\angle DAB=60°$. If the bisectors $AP$ and $BP$ of angles $A$ and $B$, respectively meets at $P$ on $CD$. Prove that $P$ is the mid-point of $CD$.

In the given figure, $ABCD$ is a square, if $\angle PQR=90°$ and $PB=QC=DR$, then prove that $QB=RC, PQ=QR$ and $\angle QPR=45°$.

In the parallelogram $ABCD$ of the given figure, $\angle PAQ$ is an obtuse angle. Two equilateral triangles $ABP$ and $ADQ$ are drawn outside the parallelogram. Prove that $△CPQ$ is also an equilateral triangle.