幾何與角

In the figure, if $\operatorname{arc}BC:\operatorname{arc}CA:\operatorname{arc}AB=1:2:3$, which of the following is/are true? I. $\angle A:\angle B:\angle C=1:2:3$ II. $a:b:c=1:2:3$ III. $\sin A:\sin B:\sin C=1:2:3$

In the figure, $TP$ and $TQ$ are tangents to the circle at $P$ and $Q$ respectively. If $M$ is a point on the minor arc $PQ$ and $\angle PMQ=\theta$, then $\angle PTQ=$

In the figure, $O$ is the centre of the circle. $AB$ touches the circle at $N$. Which of the following is/are correct? I. $M,\ N,\ K,\ O$ are concyclic. II. $\triangle HNB\sim\triangle NKB$ III. $\angle OAN=\angle NOB$

In the figure, $ABCD$ and $EFGH$ are two squares and $ACH$ is an equilateral triangle. Find $AB:EF$.

In the figure, a rectangular piece of paper $ABCD$ is folded along $EF$ so that $C$ and $A$ coincide. If $AB=12\text{ cm}$ and $BC=16\text{ cm}$, find $BE$.

In the figure, the three circles touch one another. $XY$ is their common tangent. The two larger circles are equal. If the radius of the smaller circle is $4\text{ cm}$, find the radii of the larger circles.

In the figure, $O$ is the centre of the circle. Find $\theta$.

In the figure, $ABCD$ is a square of side $6$. If $BE=CE=5$, find $AE$.

In the figure, the circle is inscribed in a regular pentagon. $P$, $Q$ and $R$ are points of contact. Find $\theta$.

In the figure, $ST$ is a tangent to the smaller circle. $ABC$ is a straight line. If $\angle TAD=2x$ and $\angle DPC=3x$, find $x$.