Circle theorems I

A, B, C, D and E are five points lying on the circumference of a circle. $EB$ is parallel to $DC$. $\angle E\hat{A}C = 72^\circ$ and $\angle A\hat{E}B = 25^\circ$. $X$ is the point where $AC$ and $EB$ intersect.

May/June 2015

In the diagram, the two circles intersect at $P$ and $Q$. The line $AB$ touches both circles at $A$ and $B$. $AD$ and $BC$ are diameters. $BD$ meets the larger circle at $R$. $\angle DBC = 40^\circ$.

May/June 2016

The circle has centre $O$ and diameter $AB$. $PA$ and $QB$ are tangents to the circle at $A$ and $B$ respectively.

May/June 2016

$A$, $B$ and $C$ lie on the circumference of a circle with centre $O$. $O$ is the midpoint of $BC$ and $\angle ABC = 38^{\circ}$. From $T$, tangents are drawn so that they touch the circle at $A$ and $B$.

May/June 2018

The diagram depicts a circle, with centre $O$, passing through $A,B,C$ and $D$. The tangents drawn at $A$ and $B$ intersect at $T$. $\angle ATB=62^\circ$ and $\angle DAB=53^\circ$.

May/June 2018

Points $A, B, C, D$ and $E$ lie on the circumference of a circle with centre $O$. The tangent $AF$ is drawn so that it touches the circle at $A$. $O$ is the midpoint of $AD$. $\angle DOE = 138^\circ$ and $\angle BEO = 12^\circ$.

May/June 2019

$A$, $B$ and $C$ lie on the circle with centre $O$, and $AB=BC$. $P$ is the midpoint of chord $AB$ while $Q$ is the midpoint of chord $BC$.

May/June 2019

The points $B,D,E,F$ and $G$ lie on the circumference of a circle whose centre is $O$. $AC$ is tangent to the circle at $B$. Angle $DFG=75^{\circ}$ and angle $ABG=48^{\circ}$.

May/June 2022

$A$, $B$, $C$ and $D$ lie on a circle with centre $O$.

May/June 2022

Points $A$ and $B$ lie on the circle with centre $O$. The lines $TA$ and $TB$ touch the circle at $A$ and $B$.

May/June 2023

$A$, $B$, $C$ and $D$ lie on the circumference of a circle with centre $O$. Angle $BCD = 58^\circ$ and angle $DBC = 72^\circ$.

May/June 2024

$A$, $B$, $C$ and $D$ lie on a circle with centre $O$. Angle $BAD = 120^{\circ}$ and angle $OBC = 20^{\circ}$.

May/June 2024

$PQRS$ forms a cyclic quadrilateral. The line $APB$ is tangent to the circle at $P$. Angle $SPB = 37^{\circ}$ and angle $PSQ = 85^{\circ}$.

May/June 2025

The points $B$, $D$, $E$ and $F$ all lie on one circle. $AC$ is a tangent to the circle at $B$. Angle $BEF = 50^{\circ}$ and $EF = BF$.

May/June 2025

In the circle with centre $O$, $AC$ is a diameter. $BCD$ and $OED$ are straight lines. $AC = 6\text{ cm}$ and $CD = 3\text{ cm}$. $\angle BAC = 34^{\circ}$.

Oct/Nov 2015

Points $A,B,C,D$ and $E$ are on circle centre $O$ in the diagram. $AD$ is a diameter, while $\angle DAC = 33^\circ$ and $\angle ACE = 70^\circ$.

Oct/Nov 2016

In the diagram, $A,B,C,D$ and $E$ are placed on a circle with centre $O$. $BOE$ is a straight line. $\angle DAB = 34^\circ$.

Oct/Nov 2016

In the diagram, $A$ and $B$ are the centres of two circles which are tangent at $P$. The line $ACT$ is tangent to the smaller circle at $T$ and cuts the larger circle at $C$. $D$ is the point on $AB$ for which $\angle CDA=90^\circ$.

Oct/Nov 2016

Points A, B, C, D and E lie on the circumference of the circle with centre $O$. $AC$ is a diameter, and $AC$ is parallel to $ED$. The lines $AC$ and $BE$ intersect at $F$. $\angle BAC$ is $52^\circ$ and $\angle CBE$ is $68^\circ$.

Oct/Nov 2017

On the circle, $A, B, C, D$ and $E$ are points. $AB$ runs parallel to $ED$. $\angle ABE = 62^\circ$, $\angle CDE = 127^\circ$ and $\angle BEC = 40^\circ$.

Oct/Nov 2018

Points $A$, $B$, $C$ and $D$ lie on the circle with centre $O$. $TA$ and $TC$ are tangents to the circle. $T$, $D$, $O$ and $B$ are on one straight line. $\angle ATO = x^\circ$.

Oct/Nov 2018

In the diagram, the points $A,B,C,D$ and $E$ are on the circle with centre $O$. The points $B,O$ and $E$ are collinear. $AB$ is parallel to $ED$ and $\angle DEO = 53^\circ$.

Oct/Nov 2019

On the circle with centre $O$, the points $A$, $B$, $C$ and $D$ are all on the circumference. Also, $\angle ACB = 69^\circ$ and $\angle DCA = 34^\circ$.

Oct/Nov 2019

$P$, $Q$, $R$ and $S$ lie on a circle. $PXR$ and $QXS$ are straight lines.

Oct/Nov 2019

The circle has centre $O$, and $AC$ and $BD$ are its diameters.

Oct/Nov 2019

The circle has centre $O$, and the points $A$, $B$, $C$, $D$ and $E$ are on its circumference. $\angle A\hat{O}B = 48^\circ$, $\angle DE\hat{B} = 54^\circ$.

Oct/Nov 2020

The points $A$, $B$ and $C$ are on a circle with centre $O$. The straight line $PBQ$ is tangent to the circle at $B$, and $O$, $C$ and $Q$ lie on one straight line. $\angle BQO = 36^\circ$ and $\angle BAC = x^\circ$.

Oct/Nov 2021

Explain why triangle $AOB$ has two equal sides and is isosceles.

Oct/Nov 2021

Points $B$, $C$ and $D$ are located on the circumference of a circle with centre $O$. $AB$ is a tangent to the circle at $B$. $BD$ is a diameter, and $OCA$ lies on a straight line. $\angle CDB = x^{\circ}$.

Oct/Nov 2022

Points $A$, $B$ and $C$ lie on the circle with centre $O$. The lines $AB$ and $AC$ are tangents to the circle. Angle $BAC = 38^\circ$.

Oct/Nov 2023

The diagram depicts a circle with centre $O$, and $AC$ and $BD$ are diameters.

Oct/Nov 2023

Determine the size of one interior angle of a regular $15$-sided polygon.

Oct/Nov 2023

Q, R, S and T lie on a circle with centre O. The line UV is tangent to the circle at T. S, O, Q and V lie on one straight line. Angle QST = 32^{\circ}.

Oct/Nov 2025

A, B, C and D lie on a circle with centre O. EF touches the circle at A, and angle BAF = $65\degree$ while angle AOD = $80\degree$.

Oct/Nov 2025