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Join 00, 00, OD;

then 00' passes through P the point of contact, (Euc. III. 12.)
and ŌC, OD are perpendicular to AB, (Euc. III. 18.)
and are therefore parallel to each other. (Euc. I. 29.)
Through D draw DE parallel to 00,

and produce OC to meet DE in E.

Then 00'DE is a parallelogram, of which the side OE is equal to the radius of the larger circle, and the adjacent side ED is equal to the sum of the given radii.

Synthesis. At the given point in the line AB,

draw CO at right angles to AB, and equal to the given radius of the smaller circle ;

produce CO to E, making OE equal to the radius of the larger circle, with center E, and radius equal to the sum of the radii of the two circles, describe a circle intersecting AB in D;

at D draw DO' at right angles to AB, and equal to the radius of the larger circle.

With center O and radius OC describe a circle, this circle will touch the given line in the given point C.

With center ✔ and radius O'D describe a circle; it will touch the line AB in D, and the other circle in the point P.

Therefore two circles are described touching each other at a point P, and each touching the given line AB, one of them at the given point C in it.

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To describe a circle which shall touch a straight line given in position, and pass through two given points.

Analysis. Let AB bẹ the given straight line, and C, D the two given points.

Suppose the circle required which passes through the points C, D to touch the line AB in the point E.

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and produce DC to meet AB in F

and let the circle be described having the center L,
join also LE, and draw LH perpendicular to CD.
Then CD is bisected in H,

and LE is perpendicular to AB.

Also, because from the point F without the circle, are drawn two straight lines, one of which FE touches the circle, and the other FDC cuts it;

therefore the rectangle contained by FC, FD, is equal to the square on FE. (111. 36.)

1

Synthesis. Join C, D, and produce CD to meet AB in F, take the point E in FB, such that the square on FE shall be equal to the rectangle FD, FC. (II. 14.)

Bisect CD in H, and draw HK perpendicular to CD;

then HK passes through the center. (III. 1, Cor. 1.)
At E draw EG perpendicular to FB,

then EG passes through the center, (III. 19,)

consequently L, the point of intersection of these two lines, is the center of the circle.

It is also manifest, that another circle may be described passing through C, D, and touching the line AB on the other side of the point F; and this circle will be equal to, greater than, or less than the other circle, according as the angle CFB is equal to, greater than, or less than the angle CFA.

PROPOSITION IV. PROBLEM.

To describe a circle passing through two given points and touching a given circle.

Analysis. Let PAB be the required circle passing through the two given points A, B, and touching the given circle whose center is O in the point P.

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At P let PQ the common tangent be drawn to the two circles, and let AB be joined and produced to meet PQ in Q.

Then the rectangle QA, QB is equal to the square on QP. From Q draw any line QCD cutting the given circle in C, D, then the rectangle QC, QD is equal to the square on QP; but the rectangle QA, QB is equal to the square on QP; therefore the rectangle QC, QD is equal to the rectangle QA, QB, wherefore the points A, B, C, D are in the circumference of a circle. Synthesis. Describe a circle passing through the given points A, B and intersecting the given circle in any points C, D.

Draw AB, CD and produce them to meet in Q.
From Q draw QP to touch the given circle in P.

Then the problem is reduced to that of describing a circle which shall pass through two given points A, B, and touch a given straight line QP at a given point P in the line. If therefore a circle ABP ‍be described by the preceding Problem passing through the two points A, B, and touching the line QP at the point P;

then the circle PAB is the circle required.

I.

5. Draw a straight line which shall touch a given circle,`and make a given angle with a given straight line.

6. Draw a tangent to an arc (of given length) of a circle, so that the length of the tangent intercepted between the extreme radii of the arc shall equal a given straight line.

7. In a given straight line find a point such that the straight line drawn from it to touch a given circle, shall be equal to a given straight line.

8. Draw a tangent to one circle the portion of which intercepted by the circumference of another circle shall equal a given straight line; provided that if the center of that circle lie without the circumference of the other, the straight line be not greater than the diameter; or if within the circumference, than the chord which touches the former circle where the line joining the centers cuts it.

9. If two tangents AB, AC be drawn to a circle, and D be any point in the circumference, the sum of the angles ABD, ACD, is

constant.

II.

10. Describe a circle which shall pass through a given point and which shall touch a given straight line in a given point.

11. Describe a circle touching a given straight line and cutting a given circle in such a way that the chord of intersection may pass at a given distance from the center of the circle.

12. Describe a circle which shall have its center in a given straight line, touch another given line, and pass through a fixed point in the first given line.

13. Describe the circles which shall pass through a given point and touch two given straight lines.

14. Show how to describe a circle that shall have its center in a given straight line, which shall pass through a given point, and also touch another given straight line.

15. Describe a circle which shall touch one given line in a given point, and shall from another given line intercept a chord of a given length.

16. Describe a circle through a given point, and touching a given straight line, so that the chord joining the given point and point of contact may cut off a segment containing a given angle.

17. Draw a circle touching a given line in a given point, and such that the angle in the segment cut off by another given line may be equal to a given angle.

18. With a given radius describe a circle which shall touch two given straight lines.

19. Describe a circle to touch two lines given in position, and such that a tangent drawn to it from a given point may be equal to a given line.

20. Describe a circle to touch two right lines given in position, so that lines drawn from a given point to the points of contact shall contain a given angle.

21. Describe a circle touching a given line at a given point, such that the tangents drawn to it from two given points in the line may be parallel.

22. A, B; are two fixed points on the circumference of a given circle, P a moveable point on the circumference, on PB is taken a point A such that PA': PA in a constant ratio; and on PA a point B', such that PB': PB in the same ratio; prove that A'B' always touches a fixed circle.

23. Describe a circle touching three given lines which are not all parallel. How many such circles are there in general? and under what circumstances are there only two such circles?

24.

III.

Describe a circle which shall have a given radius and touch a given straight line and a given circle.

25. Describe a circle with a given center, such that the circle so described and a given circle may touch one another internally.

26. To describe a circle which shall touch a given circle in a given point, and also a given straight line.

27. Describe a circle which shall have its center in a given line, and shall touch a circle and a straight line given in position.

28. Describe a circle which shall touch a given circle, have its center in a given straight line, and pass through a given point in the same straight line.

29. Find the position of a straight line, such that every two tangents drawn from the same point in this line to two given circles, may be equal.

30. A circle is drawn to touch a given circle and a given straight line; shew that the points of contact are always in the same straight line with a fixed point in the circumference of the given circle.

31. The common chord of two circles is produced to any point P; PA touches one of the circles in A, PBC is any chord of the other: shew that the circle which passes through A, B, C touches the circle to which PA is a tangent.

32. ACB being the arc of a circle, it is required to find in it a point C such that the circle described with center. C and radius CA, shall touch a given circle whose center is in B.

33. Describe a circle touching a straight line in a given point, and also touching a given circle. When the line cuts the given circle, shew that your construction will enable you to obtain six circles touching the given circle and the given line, but not necessarily in the given point.

IV.

34. Two points are given, one in each of two given circles; describe a circle passing through both points and touching one of the circles.

Draw a circle touching a given circle in a given point, and also touching another given circle.

36. To describe a circle touching a given circle in a given point, and passing through a given point not in the circumference of the given circle. In what case is this impossible?

37. Describe a circle which shall touch a given circle, and each of two given straight lines.

38. Two given circles touch each other externally, describe a circle with given radius which shall touch them both.

39. Find the semicircle which circumscribes two given circles which touch each other, and find the condition that the problem may be possible.

40. If two circles touch each other externally, describe a circle which shall touch one of them in a given point, and also touch the other. In what case does this become impossible?

41. Describe a circle touching a given straight line, and also two given circles.

42. Describe a circle which shall pass through a given point and touch a given straight line and a given circle.

43.

circles.

With a given radius to describe a circle, touching two given

44. Through a given point draw a circle touching two given circles.

45. If a circle be described touching two given circles, prove that the line joining the points of contact always passes through a given point.

46. Draw a circle which shall touch two given circles and have its center upon a given straight line.

47. One circle cuts another at the extremities of a diameter, to draw a circle touching these two circles, and having its center in the line perpendicular to the diameter at its extremity.

48. Describe a circle touching one given circle, and bisecting the circumference of another.

49. If two circles touch each other externally, and a straight line which touches both of them intersect another straight line passing through their centers, at a point whose distance from the nearer circle is equal to its diameter, the radius of one of the circles will be twice as great as that of the other.

50. The center of a given circle is equidistant from two given straight lines; to describe another circle which shall touch the two straight lines, and shall cut off from the given circle a segment containing an angle equal to a given rectilineal angle.

V.

51. To find a point P, so that tangents from it to the outsides of two equal circles which touch each other, may contain an angle equal to a given angle.

52. Given two circles; it is required to find a point from which tangents may be drawn to each, equal to two given straight lines.

53. A straight line and two circles are given. Find the point in the straight line from which tangents drawn to the circles shall be equal. 54. Find a point without two circles, such that the tangents drawn therefrom to the circles shall contain equal angles.

VI.

55. Describe three equal circles touching each other, and each passing through an angle of a given equilateral triangle.

56. With three given points, not lying in a straight line, describe three circles which shall have three common tangents.

57. Find a point from which, if straight lines be drawn to touch

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