1. If two circles touch each other externally, prove that the point of contact lies on the straight line joining their centers.

2. Prove that if two circles externally touch each other, then the point of contact lies on the straight line joining their centers.

3. Prove that if two circles touch each other, then the point of contact lies on the straight line joining their centers.

4. Prove that if two circles touch each other, then the point of contact lies on the straight line joining their centers.

5. Prove that if two circles touch each other, then the point of contact lies on the straight line connecting their centers.

6. If two circles touch each other, then the point of contact lies on the straight line joining their centers — prove it.

7. Prove that if two circles touch each other, then the point of contact lies on the straight line segment joining the centers of the two circles.

8. If two circles touch each other, then the point of contact lies on the straight line joining their centers.

9. Prove that if two circles touch externally, then the point of contact lies on the straight line connecting their centers.

10. If two circles touch each other, then the point of contact lies on the straight line joining their centers.

11. If two circles touch each other, then the point of contact lies on the straight line joining their centers.

12. Two circles intersect each other at points P and Q. If PA and PB are the diameters of the respective circles, then prove that the points A, Q, and B lie on a straight line.

13. “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.”

14. 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.

15. 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.

16. 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 \(\angle AQC = \angle BQD\).

17. Two fixed circles with centers A and B touch each other internally. Another circle touches the larger circle internally at point X and the smaller circle externally at point Y. If O is the center of this third circle, prove that AO + BO is constant.

18. 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.

19. 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.

20. Prove that the two tangents drawn to a circle from an external point are equal in length, and the line segments joining the points of contact to the external point subtend equal angles at the center of the circle.

21. 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°.

22. Two poles are placed 120 meters apart, and the height of one pole is double that of the other. If the angles of elevation to the tops of the two poles from the midpoint of the line joining their bases are complementary, find the height of the shorter pole.

23. A straight line intersects two concentric circles at points A and B on one circle and at points C and D on the other. Prove with reasoning that AC = DB.

24. 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) \(\angle BPC = \angle BQD\)

25. 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.

26. "Chords PQ and RS of a circle intersect each other at point X inside the circle. By joining P to S and R to Q, prove that triangles ∆PXS and ∆RSQ are similar. From this, prove that: PX × XQ = RX × XS Or, when two chords of a circle intersect internally, the product of the two segments of one chord is equal to the product of the two segments of the other chord.

27. 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: - \(OD = OC = OE\) - \(x - y = 4^\circ\)

28. 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.

29. Here is the English translation: > Two pillars of equal height are located directly opposite each other at points A and B on either side of a 120-meter wide road. From point C, on the line joining their bases, the angles of elevation to the tops of the pillars at A and B are 60° and 30°, respectively. Find the length of AC.

30. Two circles are touching each other externally, and the distance between their centers is 7 cm. If the radius of one circle is 4 cm, find the radius of the other circle.