Prove that a parallelogram which is not a rectangle is not cyclic.

In the picture, the circles on the left and right intersect the middle circle at $P, Q, R, S;$ the lines joining them meet the left and right circles at $A, B, C, D.$ Prove that $ABCD$ is a cyclic quadrilateral.

In the picture, points $P, Q, R$ are marked on the sides $BC, CA, AB$ of $∆ABC$ and the circumcircles of $∆AQR$ and $∆BRP$ are drawn. $\mathrm{M}$ is a point where these circles intersect. Prove that the circumcircle of $∆CPQ$ also passes through $\mathrm{M}.$

In the picture, chords $\mathrm{AB}$ and $\mathrm{CD}$ of the circle are extended to meet at $\mathrm{P}$.

Prove that the angles of $\Delta APC$ and $\Delta PBD$, formed by joining $AC$ and $BD$, are the same.

In the picture, chords $\mathrm{AB}$ and $\mathrm{CD}$ of the circle are extended to meet at $\mathrm{P}$.

Prove that $PA\times PB=PC\times PD$.

In the picture, chords $\mathrm{AB}$ and $\mathrm{CD}$ of the circle are extended to meet at $\mathrm{P}$.

Prove that if $PB=PD$, then $ABCD$ is an isosceles trapezium.