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.

Draw a rectangle of width $5$ centimetres and height $3$ centimetres.

Draw a rectangle of the same area with width $6$ centimetres.

Draw a rectangle of width $5$ centimetres and height $3$ centimetres.

Draw a square of the same area.

Draw a square of area $15$ square centimetres.

Draw a square of area $5$ square centimetres in three different ways. (Recall Pythagoras theorem)

In the picture, a line through the centre of a circle cuts a chord into two parts:

What is the radius of the circle in $\mathrm{cm}$?