5. Two circles being given, to draw a secant such that the chords intercepted by the two circles shall be of given length.

6. With the vertices of a triangle as centre, to describe three circles which shall touch one another.

7. In every right-angled triangle the diameter of the circle inscribed is equal to the excess of the sum of the two sides over the hypothenuse.

8. Construct a triangle, having given,—

Ist. The radius of the inscribed circle, one angle, and the height taken from it.

2nd. A side, the sum or difference of two other sides, and radius of the inscribed circle.

3rd. The centres of the three circles drawn to touch one side and two others produced.

9. Describe a circle which shall pass through a given point, touch a given straight line, and bisect a given circumference.

10. Describe a circle which shall pass through two given points and have its centre in a given straight line.

II. Within a given circle, place a chord of given length which shall be bisected by a given chord.

12. AB is a diameter of a circle; CD two points in the circumference on the same side of A B. Let AD, BC intersect in P and A C, B D in Q. Prove that PQ produced is perpendicular to A B.

13. Describe a circle which shall touch a given straight line in a given point, and cut off from another given straight line a chord of given length.

14. A B C is a triangle inscribed in a circle. Show that the straight line which bisects the angle A, and the straight line drawn from the centre perpendicular to B C, meet on the circumference of the circle.

15. ABC is a triangle having a right-angle at C. Find a point, D, in B C, so that the difference of the squares on A B and AD may be equal to three times the difference of the squares on A D and A ̊C.

16. Through a given point within a circle, to draw a chord which shall be bisected in that point.

17. Through a point in a circle which is not the centre, to draw the least chord.

18. To draw that diameter of a given circle which shall pass at a given distance from a given point.

19. Any two chords of a circle which cut a diameter in the same point and at equal angles, are equal to each other.

20. The two lines which join the opposite extremities of two parallel chords intersect in a point in that diameter which is perpendicular to the chords.

21. The straight lines joining towards the same parts the extremities of any two lines in a circle equally distant from the centre, are parallel to each other.

22. A, B, C, A', B', C', are points on the circumference of a circle; if the lines A B, A C be respectively parallel to A'B', A'C', show that B C' is parallel to B ̊C.

23. Two chords of a circle being given in position and magnitude, describe the circle.

24. If straight lines be drawn through the same point, terminated both ways by the circumference, they are cut less and less unequally in that point, as the angle formed with the diameter passing through that point approaches a right-angle.

25. If two equal chords of a circle cut one another either within or without a circle, the segments of the one between the point of intersection and the circumference, shall be equal to the segments of the other, each to each.

26. Of all straight lines which can be drawn from two given points to meet in the convex circumference of a given circle, the sum of those two will be the least, which make equal angles with the tangent at the point of concourse.

27. From a given point without a circle, at a distance from the circumference of the circle not greater than its diameter, draw a straight line to the concave circumference which shall be bisected by the convex circumference.

28. The diameter of a circle having been produced to a given point, it is required to find in the part produced, a point from which if a tangent be drawn to the circle, it shall be equal to the segment of the part produced, that is, the segment between the given point and the point found.

29. If an arc of a circle be divided into three equal parts by three straight lines drawn from one extremity of the arc, the angle contained by two of the straight lines is bisected by the third.

30. In a given straight line to find a point at which two other straight lines being drawn to two given points, shall contain a right-angle. Show that if the distance between the two given points be greater than the sum of their distances from the given fine, there will be two such points; if equal, there may be only one; if less, the problem may be impossible.

31. Through a given point within or without a circle, it is required to draw a straight line cutting off a segment containing a given angle.

32. Divide a circle into two parts such that the angle contained in one segment shall equal twice the angle contained in the

other.

33. Divide a circle into two segments such that the angle in one of them shall be five times the angle in the other.

34. If the diameter of a circle be one of the equal sides of an isosceles triangle, the base will be bisected by the circumference.

35. If AD, CE be drawn perpendicular to the sides BC, AB of the triangle ABC, and DE be joined, prove that the angles A DE and ACE are equal to each other.

36. If A B, CD be chords of a circle at right-angles to each other, prove that the sum of the arcs A C, BD is equal to the sum of the arcs A D, BC.

37. The sum of the arcs subtending the vertical angles made by any two chords that intersect is the same as long as the angle of intersection is the same.

38. Draw two tangents to a circle of given radius which shall contain an angle equal to a given angle.

39. If a chord of a circle be produced till the part produced be equal to the radius, and if from its extremity a line be drawn through the centre and meeting the convex and concave circumferences, the convex is one-third of the concave circumference.

40. Describe a circle the circumference of which shall pass through a given point and touch a given circle in a given point. 41. Determine the distance of a point from the centre of a given circle, so that if tangents be drawn from it to the circle, the concave part of the circumference may be double of the

convex.

42. In a chord of a circle produced, it is required to find a point from which if a straight line be drawn touching the circle, the line so drawn shall be equal to a given straight line.

43. Find a point without a given circle, such that the sum of the two lines drawn from it touching the circle, shall be equal to the line drawn from it through the centre to meet the circle.

44. Determine the point without a circle, from which, if two straight lines be drawn touching the circle, they may form an equilateral triangle with the chord which joins the points of

contact.

45. If from a point without a circle two tangents be drawn, the straight line which joins the points of contact will be bisected at right-angles by a line drawn from the centre to the point without the circle.

46. Tangents to a circle at the extremities of any chord, contain an angle which is twice the angle contained by the same chord and a diameter drawn from either of the extremities.

47. If tangents be drawn at the extremities of any two diameters of a circle, and produced to intersect one another, the straight lines joining the opposite points of intersection will both pass through the centre.

48. If any chord of a circle be produced equally both ways, and tangents to the circle be drawn on opposite sides of it from its extremities, the line joining the points of contact bisects the given chord.

Arithmetical Questions.

1. From a given point, A, a straight line is drawn, cutting a circle at points distant 3ft. and 7ft. respectively; find the length of the tangent from the same point to the circle.

Ans. 4'5825 ft. 2. The shortest distance from a point to a circle is 9 inches, and the tangent from the same point is 15 inches; find the radius of the circle.

Ans. 8in.

3. A chord is at right-angles to a diameter which it divides into parts, measuring respecting II and 5 inches; find the length of the chord. Ans. 14.8324.

4. Wishing to know the distance between an accessible object, A, and an inaccessible object, B, I raised a perpendicular, AC, and measured the distance to the point C, 10'3 yds. I then cut out in paper an angle equal to AC B. I found another point, D, such that when the paper was fixed so that along one edge I could see the point A, along the other I could see B. E was the point in a line with A and D, and also in a line with B and C. I measured A E 15 yds., DE 16 yds., CE 12 yds. From these dimensions: find the distance of the object.

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Ans. 30'297 yds.

5. A person wishing to measure the distance of an inaccessible point, O, from a station A, and having no instrument but a book, placed it at A so that on looking along one edge he saw O, and along another a point B. He walked to B and placed the book so that on looking along one edge he again saw O, and along another he saw an object in a line BC. He marked the point C which was in this line, and also in a line with A O. He then measured with the book and found AC to be three times the length of the book, and A B eight times. How many times the length of the book are the distances A O and BO?

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Ans. 21 and 22.78 times. 6. Two opposite angles, A and C, of a four-sided field, A B C D, are right-angles, and three adjacent sides are respectively D A 27 yds., AB 35 yds., and BC = 42 yds. Find the fourth side, the area of the field, and the diameter of the circle which would go through all the angular points.

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Ans. 13 784 yds.; 761.96 sqr. yds.; 27′33 yds.

CHAPTER X.

COMBINATIONS OF CIRCLES.
Concentric Circumferences.

238. Two circumferences having the same centre are said to be concentric (fig. 225).

Two concentric circumferences are everywhere at a distance from one another equal to the difference of their radii.

Fig. 225.

239. The distance of a point from a circumference is the part of the diameter passing through this point, and intercepted between it and the circumference.

When the point is exterior the theorem is evident, for let A be the point (fig. 226), and A B the diameter; draw another line A C; the straight line O BA is shorter than the bent line OCA; but the two lines OB, OC are equal; therefore A B is shorter than OC. When the point is within the circle (fig. 227),

the same construction shows that since the straight line OC or its equal O B, is shorter than the bent line O AC, the part A B is necessarily shorter than A C.

And thus the distance of a point A in one of the circumferences from the other circumference is

M