1. Sahana has drawn two circles that intersect at points P and Q. If PA and PB are the diameters of the respective circles, prove that the points A, Q, and B lie on a straight line.

2. “Two equal circles are such that the center of one lies on the other, and they intersect at points A and B. A straight line through A intersects the circles again at points C and D. Prove that triangle BCD is equilateral.”

3. Timir has drawn two circles that intersect each other at points P and Q. From point P, two straight lines are drawn: one intersects one circle at points A and B, and the other intersects the other circle at points C and D respectively. Prove that \(\angle AQC = \angle BQD\).

4. ABCD is a cyclic quadrilateral. If the sides AB and DC are extended, they meet at point P; and if the sides AD and BC are extended, they meet at point R. The circumcircles of triangles BCP and CDR intersect at point T. Prove that the points P, T, and R lie on a straight line.

5. Two circles intersect each other at points P and Q. Two straight lines passing through points P and Q intersect one circle at points A and C, and the other circle at points B and D respectively. Prove that AC is parallel to BD.

6. Two circles intersect at points A and B, and the circumference of each passes through the center of the other. If a straight line drawn through point A intersects the two circles again at points C and D respectively, then what type of triangle is BCD?

(a) right-angled isosceles (b) scalene (c) isosceles (d) equilateral

7. Two identical circles, each with radius \(r\), intersect in such a way that each circle passes through the center of the other. The centers of the circles are labeled A and B, and they intersect at points P and Q. The area of triangle \(\triangle APB\) will be:

(a) \(\cfrac{\sqrt3}{4}r^2\) (b) \(\cfrac{\sqrt3}{2}r^2\) (c) \(\cfrac{\sqrt3}{3}r^2\) (d) \(\sqrt3 r^2\)

8. A straight line intersects one of two concentric circles at points A and B, and the other at points C and D. Prove that AC = BD.

9. If AB and AC are chords of the larger of two concentric circles, and they touch the smaller circle at points P and Q respectively, prove that: \[ PQ = \frac{1}{2}BC \]

10. We have drawn two circles with centers \(A\) and \(B\), which touch each other externally at point \(C\). A point \(O\) lies on the common tangent at point \(C\), and tangents \(OD\) and \(OE\) are drawn from point \(O\) to the circles centered at \(A\) and \(B\), touching them at points \(D\) and \(E\) respectively. It is given: - \(\angle COD = 56^\circ\) - \(\angle COE = 40^\circ\) - \(\angle ACD = x^\circ\) - \(\angle BCE = y^\circ\) We are to prove: - \(OD = OC = OE\) - \(x - y = 4^\circ\)

11. I have drawn two circles with centers A and B that externally touch each other at point O. A straight line is drawn through point O, which intersects the two circles at points P and Q respectively. Prove that AP is parallel to BQ.

12. From an external point A of a circle, two tangents AB and AC are drawn, touching the circle at points B and C respectively. A tangent is drawn at point X, which lies on the arc BC, and it intersects AB and AC at points D and E respectively. Prove that the perimeter of triangle ∆ADE = 2 × AB.

13. Mohit has drawn two straight lines from an external point of a circle, which intersect the circle at points A, B and C, D respectively. Using reasoning, prove that two angles of ∆XAC and ∆XBD are equal.

14. I have drawn two circles that intersect each other at points G and H. Then I drew a straight line through point G which intersects the two circles at points P and Q. Next, I drew another straight line through point H, parallel to PQ, which intersects the two circles at points R and S. Prove that PQ = RS.

15. Two circles intersect at points A and B, and the center of one of the circles is point O. A straight line from point A intersects the circle centered at O at point R and the other circle at point P. By joining points P to B and R to B, prove that PR = PB.

16. In the adjacent figure, two circles intersect at points C and D. Two straight lines passing through points D and C intersect one circle at points A and B respectively, and the other circle at points E and F respectively. If ∠DAB = 75°, then find the value of ∠DEF.

(a) 75° (b) 70° (c) 60° (d) 105°

17. Two circles touch each other externally at point \(C\). \(AB\) is a common tangent to the two circles, touching them at points \(A\) and \(B\), respectively. The measurement of \(\angle ACB\) is –

(a) 60° (b) 45° (c) 30° (d) 90°

18. Two tangents are drawn to a circle from points A and B on the circumference, and they intersect at point C. Another point P lies on the circumference, on the side opposite to where point C is located with respect to the center. If \(\angle\)APB = 35°, then what is the measure of \(\angle\)ACB?

(a) 145° (b) 55° (c) 110° (d) None of the above

19. In the isosceles triangle ABC, AB = AC. A straight line parallel to BC intersects AB and AC at points P and Q respectively. Prove that the quadrilateral BCQP is cyclic (i.e., its vertices lie on a circle).

20. The centers of two circles are P and Q. The circles intersect at points A and B. Through point A, two straight lines parallel to the line segment PQ intersect the circles at points C and D. Prove that CD = 2PQ.

21. A straight line intersects \( \triangle PQR \) at points \( X \) and \( Y \) on \( PQ \) and \( PR \), respectively, such that \( \frac{PX}{XQ} = \frac{PY}{YR} \). If \( \angle PXY = \angle PRQ \), then prove that \( \triangle PQR \) is an isosceles triangle.

22. Draw a median \(AD\) of \(\triangle ABC\). If a straight line parallel to \(BC\) intersects \(AB\) and \(AC\) at points \(P\) and \(Q\) respectively, then prove that \(AD\) bisects the line segment \(PQ\).

23. Two circles centered at P and Q externally touch each other at point A. A common external tangent to both circles touches them at points R and S, respectively. Prove that \(\angle\)RAS = 90°.

24. In two concentric circles, the chords AB and AC of the larger circle touch the smaller circle at points P and Q respectively. Prove that PQ = \(\frac{1}{2}\)BC.

25. The centers of two concentric circles are at point O. A straight line intersects one circle at points A and B, and the other circle at points C and D. If AC = 5 cm, find the length of BD.

(a) 2.5 cm (b) 5 cm (c) 10 cm (d) none of the above

26. I have drawn two circles with centers C and D that intersect each other at points A and B. A straight line PK is drawn through point A, which intersects the circle with center C at point P and the circle with center D at point Q. Prove that: (i) \(\angle PBQ = \angle CAD\) (ii) \(\angle BPC = \angle BQD\)

27. Three points P, Q, and R lie on a circle. From point P, perpendiculars are drawn to chords PQ and PR, which intersect the circle at points S and T respectively. Prove that RQ = ST.

28. We have drawn a median \(AD\) of triangle \( \triangle ABC \). If a straight line parallel to \(BC\) intersects sides \(AB\) and \(AC\) at points \(P\) and \(Q\) respectively, then prove that the line segment \(PQ\) is bisected by the median \(AD\).

29. AB is a diameter of a circle with center O. P is any point on the circle. Tangents are drawn at points A and B, which are intersected by the tangent drawn from point P at points Q and R respectively. If the radius of the circle is \(r\), prove that \(PQ \cdot PR = r^2\).

30. Triangles \(∆ABC\) and \(∆DBC\) are drawn on the same base \(BC\) and on the same side of it. Let \(E\) be any point on side \(BC\). From point \(E\), lines parallel to \(AB\) and \(BD\) are drawn, intersecting sides \(AC\) and \(DC\) at points \(F\) and \(G\) respectively. Prove that: \(AD ∥ FG\).