**Group (2015-2019)**

**1. (2015/Myanmar /q12a )**

Two circles cut at $A, B$. The tangent to the first at $A$ meets the second again at $C$; and the tangent to the second at $B$ meets the first again at $D$. Prove that $A D$ and $C B$ are parallel. $\quad$. (5. marks)

**2. (2015/Myanmar /q12b )**

$A B C$ is a triangle in which $A B=A C$. $P$ is a point insidÃ¨ the triangle such that $\angle P A B=\angle P B C \cdot Q$ is the point on $B P$ such that $A Q=A P$. Prove that $A B C Q$ is cyclic. (5 marks )

**3. (2015/FC /q12a )**

In the figure, $A B C D E$ is a semicircle at centre $O$, the segment $A E$ is the diameter and $B, C, D$ are any points on the arc. Prove that $\angle A B C+\angle C D E=270^{\circ}$.

**4. (2015/FC /q12b )**

Given: $\angle A B E=\angle A D E$ and $\angle D A C=\angle D E C$. Prove: $A, B, C, D$ and $E$ all lie on one circle.(5 marks)

**5. (2016/Myanmar /q4 )**

Given : $\odot O$ with $A B=A D$ and $A C$ is a diameter. $$\text { Prove : } B C=C D$$

**6. (2016/Myanmar /q12a )**

Prove that the opposite angles of a quadrilateral inscribed in a circle are supplementary.

**7. (2016/Myanmar /q12b )**

$A B C D$ is a parallelogram. Any circle through $A$ and $B$ cuts $D A$ and $C B$ produced at $P$ and $Q$ respectively. Prove that $D C Q P$ is cyclic.

**8. (2016/FC /q4 )**

$A$ and $B$ are two points on a circle $3 \mathrm{~cm}$ apart. The chord $A B$ is produced to $C$ making $B C=1 \mathrm{~cm}$. Find the length of the tangent from $C$ to the circle.

**9. (2016/FC /q12a )**

In the figure, $Q P T$ is a tangent at $P$ and $P D$ is a diameter. If $\angle B P T=x, \operatorname{arc} D C=\operatorname{arc} C B$ then find $\angle D P C, \angle C P B$ and $\angle Q P C$ in terms of $x$.

**10. (2016/FC /q12b )**

Two incongruent circles $P$ and $Q$ intersect at $A$ and $D$, a line $B D C$ is drawn to cut the circle $P$ at $B$ and circle $Q$ at $C$, and such that $\angle B A C=90^{\circ}$. Prove that $A P D Q$ is cyclic.

**11. (2017/Myanmar /q4 )**

In the figure, $O$ is the centre of the circle, find $\angle R Q T$.

**12. (2017/Myanmar /q12a )**

Two unequal circles are tangent internally at $A ; B C$, a chord of the larger circle, is tangent to the smaller circle at $D ;$ prove that $A D$ bisects $\angle B A C$.

**13. (2017/FC /q4 )**

In circle $\mathrm{O}, \mathrm{PS}$ is a diameter and $\angle \mathrm{POQ}=60^{\circ}, \angle \mathrm{ROS}=70^{\circ}$. find $\angle \mathrm{PTQ} .$

**14. (2017/FC /q12a )**

In the figure, $O$ is the centre of the circle, $\mathrm{AFG} / / \mathrm{OB}, \angle \mathrm{AOB}=120^{\circ}$ and $\angle \mathrm{EAG}=80^{\circ}$.

**15. (2017/FC /q12b )**

In $\triangle \mathrm{ABC}, \mathrm{AB}=\mathrm{AC} . \mathrm{P}$ is any point on $\mathrm{BC}$, and $\mathrm{Y}$ any point on $\mathrm{AP}$. The circle BPY and CPY cut $\mathrm{AB}$ and $\mathrm{AC}$ respectively at $\mathrm{X}$ and $\mathrm{Z}$. Prove $\mathrm{XZ} / / \mathrm{BC} . \quad$ ( $5 \mathrm{marks})$

**16. (2018/Myanmar /q4 )**

In the figure, $\angle B C D=125^{\circ}$ find $x$ and $y$.

**17. (2018/Myanmar /q12a )**

Through the points of intersection of two circles, two straight lines $A B$ and $C D$ are drawn meeting one circle at $A, C$ and the other at $B, D$. Prove that $A C / / B D$.

**18. (2018/Myanmar /q12b )**

$A B C$ is a triangle inscribed in a circle and $D E$ the tangent at $A$. A line drawn parallel to $D E$ meets $A B, A C$ at $F, G$ respectively. Prove that $B F G C$ is a cyclic quadrilateral.

**19. (2018/FC /q4 )**

Given : $A B C D E F$ is an inscribed regular hexagon. $P F$ is a tangent to the circle $O$ at $F$. Prove : $P F$ and $E A$ are parallel.

**20. (2018/FC /q12a )**

If $A, B, C$ are three points on the circumference of a circle such that the chord $A B$ is equal to the chord $A C$, prove that the tangent at $A$ bisects the exterior angle between $A B$ and $A C$.

**21. (2018/FC /q12b )**

Two circles intersect at $A$ and $B$. $A$ point $P$ is taken on one so that $P A$ and $P B$ cut the other at $Q$ and $P$ respectively. The tangents at $Q$ and $R$ meet the tangent at $P$ in $S$ and $T$ respectively. Prove that $\angle T P R=\angle B R Q$ and $P B Q$. is cyclic.

**22. (2019/Myanmar /q4a )**

AT and BT are tangents to the circle $A B C$ at $\mathrm{A}$ and $\mathrm{B}$. Prove that $\angle \mathrm{BTX}=2 \angle \mathrm{ACB}$. (3 marks)

**23. (2019/Myanmar /q11a )**

PT is a tangent and PQR is a secant to a circle. A circle with $\mathrm{T}$ as centre and radius TQ. meets QR again at S. Prove that $\angle \mathrm{RTS}=\angle \mathrm{RPT} . \quad$ (5 marks)

**24. (2019/Myanmar /q12a )**

Prove that the quadrilateral formed by producing the bisectors of the interior angles of any quadrilateral is cyclic. : $\quad$ ( 5 marks)

**25. (2019/FC /q4a )**

Given: $\mathrm{XY}$ is the tangent at C. Prove: $\mathrm{XY} / / \mathrm{DE}$

**26. (2019/FC /q11a )**

$\mathrm{ABC}$ is an acute-angled triangle inscribed in a circle whose centre is $\mathrm{O}$, and $\mathrm{OD}$ is the perpendicular drawn from $\mathrm{O}$ to $\mathrm{BC}$. Prove $\angle \mathrm{BOD}=\angle \mathrm{BAC}$. $(5 \mathrm{marks})$

**27. (2019/FC /q12a )**

Two circles cut at $\mathrm{C}, \mathrm{D}$ and through $\mathrm{C}$ any line $\mathrm{ACB}$ is drawn to meet the circles at $\mathrm{A}$, B. AD and BD are joined and produced to meet the circles again at E, F. If AF, BE produced meet at $\mathrm{G}$, prove that $\mathrm{D}, \mathrm{F}, \mathrm{G}, \mathrm{E}$ are concyclic. (5 marks)

**28. (2017/Myanmar /q12b )**

$P V$ is a tangent to the circle and $Q T$ is parallel to $P V$ Prove that $Q R S T$ is a cyclic quadrilateral.

**Answer**

1. Prove

2. Prove

3. Prove

4. Prove

5. Prove

6. Prove

7. prove

8. $2 \mathrm{~cm}$

9. $\angle D P C=\frac{90-x}{2}=\angle C P B, \angle Q P C=\frac{270-x}{2}$

10. Prove

11. $\angle Q R T=62$

12. Prove

13. $65^{\circ}$

14. $60^{\circ},40^{\circ}$

15. Prove

16. $x=35, y=55$

17. Prove

18. Prove

19. Prove

20. Prove

21. Prove

22. Prove

23. Prove

24. Prove

25. Prove

26. Prove

27. Prove

28. Prove

**Group (2014)**

1. PTU is the tangent at the point $T$ to the circle and $P Q R$ is a straight line. If $\angle P Q T=100^{\circ}, \angle U T S=55^{\circ}$ and $P Q=Q T$, find $\angle T S R, \angle T R Q$ and $\angle S T R .$ (5 marks)

2. In the figure, $O$ is the centre of the circle and $\angle A O Q=90^{\circ}$. Prove that $\angle O P A=\angle O Q B$. (5 marks)

3. $A C$ and $B D$ are chords of a circle. Given that $\angle B P C=72^{\circ}, \angle P C D=18^{\prime}$ and $C P=C B$, find $\angle P D C, \angle A B P$ and show that $A C$ is a diameter. (5 marks)

4. In the figure, $O$ is the centre of the circle. $A C=C B$ and $A E$ is the tangent at $A$ which meets $B D$ produced at $E$. Given that $\angle E A D=32^{\circ}$, calculate $\angle B O C$ and $\angle A E D .$ (5 marks)

5. OA and $O B$ are two radii in $\odot O$. In the figure, if $A C / / B D$ and $\angle A O B=120^{\circ}$, prove that $\triangle A E C$ is equilateral. (5 marks)

6. $A B C D$ is a quadrilateral inscribed in a circle whose centre is $O . A B$ is a diameter of the circle. If $B C=C D$, prove that $\angle B D C=\angle C A D$, $\angle B O D=4 \angle C A D$ and $\angle A B D+2 \angle D B C=90^{\circ}$. (5 marks)

7. Two circles intersect at $A$ and $B$. At A a tangent is drawn to each circle meeting the circles again at $P$ and $Q$ respectively. Prove that $\angle A B P=\angle A B Q$ and $A B^{2}=B P \cdot B Q$. (5 marks)

8. Two circles intersect at $P$ and $Q .$ At $P$, a tangent is drawn to each circle meeting the circles again at $X$ and $Y$ respectively. Prove that $\angle P Q X=\angle P Q Y$ and $P Q^{2}=Q X \cdot Q Y$. (5 marks)

9. $A T$ is a tangent at $T$ and $A B C$ is a secant to a circle. A circle with $T$ as center and radius $T B$ meets $B C$ again at $D .$ Prove that $\angle C T D=\angle C A T$. (5 marks)

10. A triangle is inscribed in a circle. Prove that the sum of the measures of the angles in the segments exterior to the triangle is four right angles. (5 marks)

11. A circle passes through the vertex $A$ of an equilateral triangle $A B C$ and is tangent to $B C$ at its midpoint $D$. Find $A E: E C .$ (5 marks)

12. In the figure, $F G$ is a diameter and $H I$ is the tangent at $G$. Prove that $L, H, I, M$ are concyclic. (5 marks)

13. In the figure, $A B$ is a diameter and $C D$ is the tangent at $B$. Prove that $\frac{A C}{A D}=\frac{A F}{A E}$. (5 marks)

14. $A B$ is a diameter of a circle and $C D$ is a tangent at $B \cdot A C$ and $A D$ cut the circle at $G$ and $H$ respectively. Prove that $A C \cdot A G=A D \cdot A H$. (5 marks)

15. In $\triangle A B C, A X, B Y, C Z$ are the perpendiculars from the vertices to the opposite sides. If the perpendiculars meets at $O$, prove that $A O \cdot O X=B O \cdot O Y=C O \cdot O Z$. (5 marks)

16. $A B C$ is a triangle inscribed in a circle. The tangent at A meets $C B$ produced at $D$ and $A E$ bisecting $\angle C A B$ meets side $C B$ at $E$. Prove that $\triangle D A E$ is isosceles. (5 marks)

17. Through the points of intersection of two circles two straight lines $A B$ and $C D$ are drawn meeting one circle at $A, C$ and the other at $B, D$. Prove that $A C$ parallel to $B D$. (5 marks)

18. Two chords $A B$ and $C D$ of a circle intersect at right-angles at $K, E$ is the midpoint of $K D$. If $A K=6 \mathrm{~cm}, C K=3 \mathrm{~cm}$ and $K D=4 \mathrm{~cm}$, find the length of $B E$. If $A E$ is produced to meet the circle again at $F$, show that $A E=4 E F .$ (5 marks)

19. $O$ is a point inside $\triangle A B C, B O, C O$ produced meet $A C, A B$ and $X, Y$ respectively. If $A X O Y$ and $B Y X C$ are cyclic quadrilateral, prove that $B X$ and $C Y$ are altitude of $\triangle A B C .$ (5 marks)

**Answer**

1. $\angle T S R=100^{\circ}, \angle T R Q=40^{\circ}, \angle S T R=25^{\circ} \quad$

2. Prove

3. $\angle A B P=18^{\circ}, \angle P D C=54$

4. $\angle B O C=58^{\circ}, \angle A E D=26^{\circ}$

5. Prove

6. Prove

7. Prove

8. Prove

9. Prove

10. Prove

11. $3: 1$

12. Prove

13. Prove

14. Prove

15. Prove

16. Prove

17. Prove

18. Prove

19. Prove

**Group (2013)**

1. Given that $M N O P Q R$ is a hexagon inscribed in a circle, show that $\angle R M N+\angle N O P+\angle P Q R=360^{\circ}$. (5 marks)

2. $A, B$ and $C$ are three points on the circumference of a circle such that $A B=A C$. Prove that the tangent at $A$ bisects the exterior angle between $A B$ and $A C$. (5 marks)

3. If $P, Q, R$ are three points on the circumference of a circle such that the chords $P Q$ is equal to the chord $P R$, prove that the tangent at $P$ bisects the exterior angle between $P Q$ and $P R$. (5 marks)

4. Two unequal circles are tangent externally at $O \cdot A B$ is a chord of the first circle. $A B$ is tangent to the second circle at $C$, and $A O$ meets this circle at $E$. Prove that $\angle B O C=\angle C O E$. (5 marks)

5. $P Q R$ is an acute triangle, a circle is described on the side $Q R$ as a diameter and cuts $P Q$ and $P R$ in $X$ and $Y$ respectively. $Q Y$ and $R X$ intersect in $Z$. Prove that $\angle Q Z R=\angle P Q R+\angle P R Q$. (5 marks)

6. Two unequal circles are tangent internally at $A ; B C$, a chord of the larger circle, is tangent to the smaller circle at $D$; prove that $A D$ bisects $\angle B A C$. (5 marks)

7. Two circles $P$ and $Q$ intersect at $A$ and $D$, two parallel lines $B A C$ and $E D F$ are drawn to cut the circle $P$ at $B, E$ and circle $Q$ at $C, F$ respectively. Show that $B C F E$ is a parallelogram. (5 marks)

8. $P Q R$ is an acute triangle inscribed in a circle whose centre is $O$, and $O S$ is the perpendicular drawn from $O$ to $Q R$. Prove that $\angle Q O S=\angle Q P R$. (5 marks)

9. (Fig)Given $\odot O$ with diameter $C I$, $C M \| O N$, prove that arc $M N=\operatorname{arc} N I$. (5 marks)

10. Two incongruent circles $P$ and $Q$ intersect at $A$ and $D$, a line $B D C$ is drawn to cut the circle $P$ at $B$ and circle $Q$ at $C$, and such that $\angle B A C=90^{\circ}$. Prove that A, $P, D, Q$ are concyclic. (5 marks)

11. Two incongruent circles $P$ and $Q$ intersect at $A$ and $D$, a line $B D C$ is drawn to cut the circle $P$ at $B$ and circle $Q$ at $C$, and such that $\angle B A C=90^{\circ}$. Prove that $A P D Q$ is cyclic. (5 marks)

12. Two unequal circles intersect at $P$ and $Q$ with their centres on opposite sides of the common chord $P Q$. Through $P$ the diameters $P A$ and $P B$ are drawn. The tangents at $A$ and $B$ meet at $C$. Prove that $A Q B$ is a straight line. Prove also that a circle can be drawn through the points $A, P, B$ and $C$. (5 marks)

13. Two circles intersect at $A$ and $B$. A point $P$ is taken on one so that $P A$ and $P B$ cut the other at $Q$ and $R$ respectively. The tangents at $Q$ and $R$ meet the tangent at $P$ in $S$ and $T$ respectively. Prove that $\angle T P R=\angle B R Q$ and $P B Q S$ is cyclic. (5 marks)

14. $P Q R S$ is a parallelogram. A circle through $P, Q$ cuts the diagonals $P R, Q S$ at $A$, $B$ respectively. Prove that $A, B, S, R$ are concyclic. (5 marks)

15. $A B C D$ is a parallelogram. Any circle through $A$ and $B$ cuts $D A$ at $P$ and $C B$ produced at $Q$. Prove that $D C Q P$ is cyclic. (5 marks)

16. In $\triangle A B C, L, M, N$ are the midpoints of the sides $A B, B C, A C$ respectively. If $A D \perp B C$, show that $D, L, M, N$ are concyclic. (5 marks)

17. (Fig)In the figure, $X$ is the mid-point of the chord $A B$ and $X Y$ is parallel to $A T$, the tangent at $A$. Prove that $\angle A Y X=\angle A B C$ and $B X Y C$ is a cyclic quadrilateral. Prove also that $A B^{2}=2 A Y A C$. (5 marks)

18. (Fig)Circles $P$ and $Q$ are congruent and tangent externally at $O$. Prove that $O A=O B$. (5 marks)

19. $A B C$ is Ã¡ triangle in which $A B=A C . P$ is a point inside the triangle such that $\angle P A B=\angle P B C$. Given that $Q$ is a point on $B P$ produced such that $A B C Q$ is a cyclic quadrilateral, prove that $A Q=A P$. (5 marks)

20. $A B$ is a diameter of a circle and $E$ any point on the circumference. From any point $C$ on $A B$ produced, a line is drawn perpendicular $A B$, meeting $A E$ produced at $D$. Prove that $A E \cdot A D=A B \cdot A C$. (5 marks)

21. A circle passes through the vertex $A$ of an equilateral triangle $A B C$ and is tangent to $B C$ at its midpoint $D($, cuts $A C$ at $E)$. Find $A E: E C$. (5 marks)

**Answer (2013)**

21. 3:1

#### ** Group (2012)**

$\quad$ | $\,$ | |
---|---|---|

1. | $A B C$ is a triangle inscribed in a circle. The tangent at $A$ meets $C B$ produced at $T$ and $P$ is a point on $B C$ such that $T A=T P$. Prove that $\angle B A P=\angle C A P$. (5 marks) | |

2. | $P Q R$ is a triangle inscribed in a circle. The tangent at $P$ meets $R Q$ produced at $T$, and $P C$ bisecting $\angle R P Q$ meets side $R Q$ at $C$. Prove $\Delta T P C$ isosceles. (5 marks) | |

3. | In $\triangle A B C, A B=A C . P$ is any point on $B C$, and $Y$ is any point on $A P$. The circles $B P Y$ and $C P Y$ cut $A B$ and $A C$ respectively at $X$ and $Z$. Prove $X Z / / B C$. (5 marks) | |

4. | $P Q R S$ is a cyclic quadrilateral, $Q R$ and $P S$ are produced to meet at $E$. If $\angle E S R=\angle R P Q$, then.prove that $Q R=P R$ and $\angle Q S R=\angle R S E .$ (5 marks) | |

5. | $P Q R$ is a triangle inscribed in a circle and $A B$ is the tangent at $P .$ A line $C D$ is drawn to meet $P Q, P R$ at $C, D$ respectively. If $C D R Q$ is cyclic, show that $A B / / C D$. (5 marks) | |

6. | $A B C$ is a triangle in which $A B=A C . P$ is a point inside the triangle such that $\angle P A B=\angle P B C . Q$ is the point on $B P$ such that $A Q=A P$. Prove that $A B C Q$ is cyclic. (5 marks) | |

7. | In $\triangle P Q R, X, Y, Z$ are the middle points of $P Q, P R, Q R$ respectively, and $O$ is the point of perpendicular from $Q$ to $P R$. Then prove that $X, Z, Y, O$ are concylic. (5 marks) | |

8. | In a parallelogram $A B C D, A M \perp B C$ and $C N \perp A B$. If $A M$ and $C N$ intersect at $E$, show that $A, C, D$ and $E$ are concyclic. (5 marks) | |

9. | If $L, M, N$ be the middle points of the sides of a $\triangle A B C$, and it $P, Q, R$ be the feet of the perpendiculars from the vertices on the opposite sides, prove $P, N, Q, L, M, R$, are concyclic. (5 marks) | |

10. | $A B C D$ is a square and $E$ the middle point of $C D$. A circle drown through $A, B$ and $E$ meets $B C$ at $F$. Prove $C F=\frac{1}{4} C B$. (5 marks) | |

11. | $A B C D$ is a square and $E$ the middle point of $C D$. A circle drawn through $A, B$ and $E$ meets $B C$ at $F$. Prove that $B C=4 C F$. (5 marks) | |

12. | The line $T C B$ cuts a circle at $C$ and $B$ and the line $T A$ is a tangent to the circle at $A$. Given that $A B=A T$ prove that $C A=C T$. Given also that $B C$ is a diameter of the circle, calculate $\angle A T C$. (5 marks) | |

13. | $P, Q, R, S$ are points on a circle. $P R$ and $Q S$ cross at $X$, if $P X=10 \mathrm{~cm}, S X=4 \mathrm{~cm}$, $R S=2.5 \mathrm{~cm}$, then find $P Q$. (5 marks) | |

14. | $P Q$ is a chord of a circle and $R$ is any point on the major arc $P Q$. Two chords $P Q$. and $R S$ intersect at $A$. The circle tangent to $P Q$ at $P$ and passes through $R$ cuts $R S$ at $B$. If $A B=A S$, then show that $P A=A Q$. (5 marks) | |

15. | Through the points of intersection of two circles two straight lines $A B$ and $C D$ are drawn meeting one circle at $A, C$ and the other at $B, D$. Prove that $A C \| / B D$. (5 marks) | |

16. | Iwo circle cut at $A, B$. The tangent to the first at $A$ meets the second again at $C$; and the tangent to the second at $B$ meets the first again at $D .$ Prove that $A D$ and $C B$ are parallel. (5 marks) | |

17. | In the figure $Q P T$ is a tangent at $P$ and $P D$ is a diameter. If $\angle B P T=x, \operatorname{arc} D C=\operatorname{arc} C B$ then find $\angle D P C$ and $\angle C P B$ and $\angle Q P C$ in terms of $x$. (5 marks) | |

18. | The tangent at the point $C$ on a circle meets the diameter $A B$ produced at $T$. If $\angle B C T=27^{\circ}$, calculate $\angle C T A$. If $C T=t$ and $B T=x$, prove that the radius of the circle is $\frac{t^{2}-x^{2}}{2 x}$. (5 marks) |

#### ** Answer (2012)**

$\quad$ | $\,$ | |
---|---|---|

1. | Prove | |

2. | Prove | |

3. | Prove | |

4. | Prove | |

5. | Prove | |

6. | Prove | |

7. | Prove | |

8. | Prove | |

9. | Prove | |

10. | Prove | |

11. | Prove | |

12. | $30^{\circ}$ | |

13. | $6 .25 \mathrm{~cm}$ | |

14. | Prove | |

15. | Prove | |

16. | Prove | |

17. | $\frac{90^{\circ}-x}{2}, \frac{90^{\circ}-x}{2}, \frac{270^{\circ}-x}{2}$ | |

18. | $36^{\circ}$ |

## ** Group (2011)**

$\quad\;\,$ | $\,$ | |
---|---|---|

1. | In figure if $A P=10, P D=6$, $D A=12, B C=9$, find $A B$ and $C D$. $\mbox{ (5 marks)}$ | |

2. | In the figure $A C$ is tangent to the $\odot A B D$; $C B D$ and $D A E$ are straight lines. Find $B D$ and $A E$ using the given data in the figure. $\mbox{ (5 marks)}$ | |

3. | $A B C$ is a triangle inscribed in a circle whose centre is $O$, and $O D$ is the perpendicular drawn from $O$ to $B C$.Prove that $\angle B O D=\angle B A C$. $\mbox{ (5 marks)}$ | |

4. | Two circles intersect at $A, B$.At A a tangent is drawn to each circle meeting the circles again at $P$ and $Q$ respectively.Prove that $\angle A B P=\angle A B Q$. $\mbox{ (5 marks)}$ | |

5. | Two circles intersect at $P ; Q \cdot$ At $P$, a tangent is drawn to each circle meeting the circles again at $R$ and $S$ respectively.Prove that $\angle P Q R=\angle P Q S$. $\mbox{ (5 marks)}$ | |

6. | $O A$ and $O B$ are two radii of a circle meeting at right angles.From $A, B$ two parallel chords $A X, B Y$ are drawn.Prove that $A Y \perp B X$. $\mbox{ (5 marks)}$ | |

7. | In the figure $A T$ is a tangent at $A$ and $A C$ is a chord.If $B$ is the middle point of arc $A B C$, prove that $A B$ bisects $\angle C A T$, and the perpendiculars from $B$ to the tangent and the chord are equal. $\mbox{ (5 marks)}$ | |

8. | Two circles intersect at $A$ and $B$.At $A$ a tangent is drawn to each circle meetin\xi the circles again at $P$ and $Q$ respectively. Prove that $\angle A B P=\angle A B Q$.Prove alsc that $A B^{2}=B P \cdot B Q$. $\mbox{ (5 marks)}$ | |

9. | $A B C$ is a triangle inscribed in a circle and $D E$ the tangent at $A$.A line drawn parallel to $D E$ meets $A B, A C$ at $F, G$ respectively.Prove that $B F G C$ is a cyclic quadrilateral. $\mbox{ (5 marks)}$ | |

10. | Two circles cut at $A$ and $B$.Through $A$ a line $C A D$ is drawn to meet the circles at $C$ and $D. C B$ and $D B$ are joined and produced to meet the circles again at $E$ and $F$. If $C F$ produced and $D E$ produced meet at $G$, prove that the points $B, F, G, E$ are concyclic. $\mbox{ (5 marks)}$ | |

11. | $A B C D$ is a parallelogram.A circle through $A, B$ cuts $B C, A C, B D$ and $A D$ at $H, Q, P, K.$ Prove that $C, D, H, K$ are concyclic. $\mbox{ (5 marks)}$ | |

12. | Given: $\angle A B E=\angle A D E$ and $\angle D A C=\angle D E C$ Prove: $A, B, C, D$ and $E \mathrm{a}^{1}$ llie on one circle. $\mbox{ (5 marks)}$ | |

13. | From a point $D$ on the base $B C$ of $\triangle A B C$ a line is drawn meeting $A B$ at $E$ and such that $\angle B D E=\angle A.$ Prove that $B E \cdot B A=B D \cdot B C$. $\mbox{ (5 marks)}$ | |

14. | $\triangle P Q R$ is a triangle in which $P L, Q M$ and $R N$ are the perpendiculars drawn from the vertices to the opposite sides.If the perpendiculars meet at $O$, prove that $P O \cdot O L=Q O \cdot O M=R O \cdot O N$. $\mbox{ (5 marks)}$ | |

15. | In the figure, $A B$ is a diameter and $C D$ is the tangent at $B$.Prove that $\frac{A C}{A D}=\frac{A F}{A E}$. $\mbox{ (5 marks)}$ |

#### ** Answer (2011)**

$\quad\;\,$ | $\,$ | |
---|---|---|

1. | $\mathrm{AB}=14.5, \mathrm{CD}=13.5$ | |

2. | $\mathrm{BD}=2.2, \mathrm{AE}=4.2$ | |

3. | Prove | |

4. | Prove | |

5. | Prove | |

6. | Prove | |

7. | Prove | |

8. | Prove | |

9. | Prove | |

10. | Prove | |

11. | Prove | |

12. | Prove | |

13. | Prove | |

14. | Prove | |

15. | Prove |

## ** Group (2010)**

$\quad\;\,$ | $\,$ | |
---|---|---|

1. | (figure)Given $\odot O$ with $A C=B D$, prove that $\triangle P B C$ is isosceles with base $B C.$ $\text{ (5 marks)}$ | |

2. | In a circle with centre $S$, $AB$ and $BC$ are congruent chords. $SV$ and $SU$ are two radii such that $S V \perp A B$ and $S U \perp B C.$ Prove that $B$ is the midpoint of arc $V U.$ $\text{ (5 marks)}$ | |

3. | (figure)Circles $P$ and $Q$ are congruent and tangent externally at $O.$ Prove that $O A=O B.$ $\text{ (5 marks)}$ | |

4. | Two circles intersect at $M, N$ and from $M$ diameters $M A, M B$ are drawn in each circle.If $A, B$ be joined to $N$, prove that $A N B$ is a straight line. $\text{ (5 marks)}$ | |

5. | (figure)Given $\odot O$ with diameter $C I$, $C M \| O N$, prove that $\operatorname{arc} M N=\operatorname{arc} N I.$ $\text{ (5 marks)}$ | |

6. | (figure)Given $\odot G$ with diameter $C I$, $C M \| G N$, prove that $\operatorname{arc} M N=\operatorname{arc} N I.$ $\text{ (5 marks)}$ | |

7. | Draw a circle and a tangent, $T A S$ meeting it at $A.$ Draw a chord $A B$ making $\angle T A B=60^{\circ}$ and another chord $B C \| T S.$ Prove that $\triangle A B C$ is equilateral. $\text{ (5 marks)}$ | |

8. | Draw a circle and a tangent $T A S$ meeting it at $A.$ Draw a chord $A B$ making $\angle T A B=30^{\circ}$ and another chord $B C \| T S.$ Prove that $A B=A C$ and find $\angle B A C.$ $\text{ (5 marks)}$ | |

9. | $P Q R S$ is a square.$A$ circle through $P, Q$ touches $R S$ at $M$ and cuts $Q R$ at $N$, such that $N R=\frac{1}{4} Q R.$ Prove that $M$ is the middle point of $R S.$ $\text{ (5 marks)}$ | |

10. | $A B C$ is a triangle inscribed in a circle.The tangent at $A$ meets $C B$ produced at $D$, and $A E$ bisecting $\angle C A B$ meets side $C B$ at $E.$ Prove that $\triangle D A E$ is isosceles. $\text{ (5 marks)}$ | |

11. | $T A$ and $T B$ are tangents to the circle with centre at $O$ and $T A$ produced meets $B O$ produced at $X$ and $X B=4 \mathrm{~cm}, T B=3 \mathrm{~cm}.$ Find $T A, T X$ and radius of the circle. $\text{ (5 marks)}$ | |

12. | (figure)In the figure $\angle B A C=40^{\circ}, \angle A F D=85^{\circ}$ and $\angle F D C=30^{\circ}$ Prove that $B C E F$ is a cyclic quadrilateral. $\text{ (5 marks)}$ | |

13. | (figure)In the figure, $\angle E A C=40^{\circ}, \angle A E D=80^{\circ}$, $\angle C D E=20^{\circ}.$ Prove that $B, C, F, E$ are concyclic.$\angle C D E=$ $A$. $\text{ (5 marks)}$ | |

14. | (figure)In the figure, $\angle E A F=45^{\circ}, \angle A E D=80^{\circ}$, $\angle C D E=25^{\circ}.$ Prove that $B, C, F, E$ are concyclic. $\text{ (5 marks)}$ | |

15. | $A B$ is a diameter of the circle $A P B.$ A line perpendicular to $A B$ intersects $A B$, $A P$ at $H, K$ respectively.Prove that $K H B P$ is cyclic. $\text{ (5 marks)}$ | |

16. | $P Q R S$ is a parallelogram.A circle through $P, Q$ cuts the diagonals $P R, Q S$ at $A, B$ respectively.Prove that $A, B, \mathrm{~S}, R$ are concyclic. $\text{ (5 marks)}$ | |

17. | In circle $O$, the diameter $A B$ is produced to $C$ and the line $C T$ is a tangent to the circle at $T.$ The line drawn perpendicular to $A C$ at $C$ meets $A T$ produced at $D.$ Prove that $B C D T$ is cyclic quadrilateral and $C T=C D.$ $\text{ (5 marks)}$ | |

18. | Prove that the quadrilateral formed by producing the bisectors of the interior angles of any quadrilateral is cyclic. $\text{ (5 marks)}$ | |

19. | (figure)In the figure, $O$ is the centre of the circle.The diameter $A B$ is produced to $C$ and the line $C T$ is the tangent to the circle at $T.$ The line drawn perpendicular to $A C$, at $C$, meets $A T$ produced at $D.$ Prove (i) $B C D T$ is a cyclic quadrilateral. $\text{ (5 marks)}$ | |

20. | Two circles $P$ and $Q$ intersect at $A$ and $D.$ A line $B D C$ is drawn to cut the circle $P$ at $B$ and the circle $Q$ at $C.$ If $A P D Q$ is a cyclic quadrilateral, prove that $B C^{2}=A B^{2}+A C^{2}$. $\text{ (5 marks)}$ |

#### ** Answer (2010)**

$\quad\;\,$ | $\,$ | |
---|---|---|

1. | Prove | |

2. | Prove | |

3. | Prove | |

4. | Prove | |

5. | Prove | |

6. | Prove | |

7. | Prove | |

8. | $120^{\circ}$ | |

9. | Prove | |

10. | Prove | |

11. | TA=3 cm,TX=5 cm, radius=1.5 cm | |

12. | Prove | |

13. | Prove | |

14. | Prove | |

15. | Prove | |

16. | Prove | |

17. | Prove | |

18. | Prove | |

19. | Prove | |

20. | Prove |

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