The rectangle circumscribed about a circle is a square. Prove it.

$ABC$ is an isosceles triangle of which $AB=AC$. A circle inscribed in the triangle touches the sides $AB, BC$ and $CA$ at $D, E$ and $F$ respectively. Prove that $BE=CE$.

From an external point $A$, a tangent is drawn to a circle with centre $O$ and the tangent touches the circle at $B$. If $OB=9 \mathrm{cm}, AO=15 \mathrm{cm}$, then the length of $AB$ is

The radius of two circles are $9 \mathrm{cm}$ and $5 \mathrm{cm}$. The circles touch externally. The distance between the centres is

Two circles touch externally at $A$. A common tangent $PQ$ to two circles touches the circles at $P$ and $Q$. Then the measure of $\angle PAQ$ is

The radius of two circles are $4.5 \mathrm{cm}$ and $2 \mathrm{cm}$. The two circles touch internally. The distance between the centres is

The radius of a circle with centre $O$ is $6 \mathrm{cm}$. $P$ is a point at a distance $10 \mathrm{cm}$ from $O$. $PA$ and $PB$ are two tangents from $P$ to the circle. Then the area of the quadrilateral $APBO$ is

Two circles of radii $x$ and $y$ touch each other externally. The distance between their centres is _____.

Two circles of radii $a$ and $b\left(<a\right)$ touch each other internally. The distance between their centres is _____.