幾何與角

A circle, centre $O$, touches the sector $ABC$ internally at $D$, $E$ and $F$. $\angle C=60^\circ$ and $AC=18$. Find the radius of the circle.

In the figure, $PQ$ is a diameter and $PT$ is a tangent of the circle. $QT$ cuts the circle at $R$. Let $\angle Q=\theta$ and $PQ=x$, then $TR=$

In the figure, $ABCD$ is a cyclic quadrilateral. $BA$ is produced to $E$. $DA$ bisects $\angle CAE$. $\angle BCD=$

The exterior angles of a pentagon are $x^\circ$, $2x^\circ$, $3x^\circ$, $4x^\circ$, $5x^\circ$. The smallest interior angle of the pentagon is

In the figure, $A$, $D$, $E$ and $B$ lie on a straight line. $CE$ bisects $\angle ACB$ and $CD\perp AB$. $\angle DCE=$

In the figure, $ABCD$ is a rhombus. $B$ is the centre of the circle. $\angle ABC=$

In the figure, $AB$ is a diameter of the circle $ABC$. If arc $AC$ has the same length as $AB$, then $\angle CAB=$

In the figure, $ABCD$ is a rectangle. $E$ is a point on $BC$ such that $\angle AED=90^\circ$. $AD=13$ and $DE=5$. The area of $ABCD=$

In the figure, $ABCD$ is a rectangle. $E$, $F$, $G$ and $H$ are points on the four sides such that $EF\parallel DB\parallel GH$. $AF=FB$ and $HC=2BH$. What fraction of the area of $ABCD$ is shaded?

In the figure, $FG$ touches the circle at $E$. The chord $CB$ is produced to meet $FG$ at $A$. $\angle ACE=$