Circle theorems II

On the diagram, $A,B,C,D$ are on the circle with centre $O$. $AD$ is a diameter. The tangent at $B$ to the circle intersects the extension of line $DA$ at $T$. $\angle A\hat{O}B=68^\circ$ and $\angle C\hat{A}O=43^\circ$.

May/June 2015

In the diagram, $P, Q, R, S$ and $T$ are points on the circle. $QT$ is a diameter of the circle with centre $O$. $X$ is the point where $PS$ and $QT$ cross. $\angle PXT = 125^\circ$ and $\angle PSQ = 35^\circ$.

May/June 2017

$PQ$ runs parallel to $RS$. $ABCD$ lies on a straight line. Also, $BE = CE$ and $\angle ABE = 110^\circ$. Points $U, V, W, X$ and $Y$ are located on the circumference of a circle with centre $O$. $UY$ is a diameter of the circle, and $ZX$ is tangent to the circle at $X$. $\angle VUX = 35^\circ$, $\angle XZY = a^\circ$ and $\angle VWY = b^\circ$.

May/June 2023

The circles centred at $O$ and $Q$ are shown touching at $P$, with $OPQ$ lying on one straight line. The straight line $ORT$ cuts the smaller circle at $R$ and is a tangent to the larger circle at $T$. $OR = 4\,\text{cm}$ and $RT = 6\,\text{cm}$. The radius of the larger circle is $x\,\text{cm}$.

Oct/Nov 2020

$ABD$ forms a triangle. $C$ lies on $BD$, and $E$ lies on $AD$. $AB$ runs parallel to $EC$, and $CD = DE$.

Oct/Nov 2024