wherefore the squares on AE, EC, DE, EB, are equal to the square on AF, the diameter of the circle. When the chords meet without the circle, the property is proved in a similar manner. I. 7. THROUGH a given point within a circle, to draw a chord which shall be bisected in that point, and prove it to be the least. 8. To draw that diameter of a given circle which shall pass at a given distance from a given point. 9. Find the locus of the middle points of any system of parallel chords in a circle. 10. The two straight lines which join the opposite extremities of two parallel chords, intersect in a point in that diameter which is perpendicular to the chords. 11. The straight lines joining towards the same parts, the extremities of any two lines in a circle equally distant from the center, are parallel to each other. 12. A, B, C, A', B', C are points on the circumference of a circle; if the lines AB, AC be respectively parallel to A'B', A'C', shew that BC' is parallel to B'C. 13. Two chords of a circle being given in position and magnitude, describe the circle. 14. Two circles are drawn, one lying within the other; prove that no chord to the outer circle can be bisected in the point in which it touches the inner, unless the circles are concentric, or the chord be perpendicular to the common diameter. If the circles have the same center, shew that every chord which touches the inner circle is bisected in the point of contact. 15. Draw a chord in a circle, so that it may be double of its perpendicular distance from the center. 16. The arcs intercepted between any two parallel chords in a circle are equal. 17. If any point P be taken in the plane of a circle, and PA, PB, PC,..be drawn to any number of points A, B, C,..situated symmetrically in the circumference, the sum of PA, PB,..is least when P is at the center of the circle. II. 18. 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. 19. From a point without a circle two straight lines are drawn cutting the convex and concave circumferences, and also respectively parallel to two radii of the circle. Prove that the difference of the concave and convex arcs intercepted by the cutting lines, is equal to twice the arc intercepted by the radii. 20. In a circle with center O, any two chords, AB, CD are drawn cutting in E, and OA, OB, OC, OD are joined; prove that the angles AOC BOD=2.AEC, and AOD + BOC=2.AÊD. 21. If from any point without a circle, lines be drawn cutting the circle and making equal angles with the longest line, they will cut off equal segments. 22. If the corresponding extremities of two intersecting chords of a circle be joined, the triangles thus formed will be equiangular. 23. 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. 24. If on two lines containing an angle, segments of circles be described containing angles equal to it, the lines produced will touch the segments. 25. Any segment of a circle being described on the base of a triangle; to describe on the other sides segments similar to that on the base. 26. 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. 27. If the chord of a given circular segment be produced to a fixed point, describe upon it when so produced a segment of a circle which shall be similar to the given segment, and shew that the two segments have a common tangent. 28. If AD, CE be drawn perpendicular to the sides BC, AB of the triangle ABC, and DE be joined, prove that the angles ADE, and ACE are equal to each other. 29. If from any point in a circular arc, perpendiculars be let fall on its bounding radii, the distance of their feet is invariable. III. 30. If both tangents be drawn, (fig. Euc. III. 17.) and the points of contact joined by a straight line which cuts EA in H, and on HA as diameter a circle be described, the lines drawn through E to touch this circle will meet it on the circumference of the given circle. 31. Draw, (1) perpendicular, (2) parallel to a given line, a line touching a given circle. 32. If two straight lines intersect, the centers of all circles that can be inscribed between them, lie in two lines at right angles to each other. 33. Draw two tangents to a given circle, which shall contain an angle equal to a given rectilineal angle. 34. Describe a circle with a given radius touching a given line, and so that the tangents drawn to it from two given points in this line may be parallel, and shew that if the radius vary, the locus of the centers of the circles so described is a circle. 35. Determine the distance of a point from the center 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. 36. 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. 37. 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 center to meet the circle. 38. 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 center to the point without the circle. 39. 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 center. 40. If from any point without a circle two lines be drawn touching the circle, and from the extremities of any diameter, lines be drawn to the point of contact cutting each other within the circle, the line drawn from the points without the circle to the point of intersection, shall be perpendicular to the diameter. 41. 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. 42. AB is a chord, and AD is a tangent to a circle at A. DPQ any secant parallel to AB meeting the circle in P and Q. Shew that the triangle PAD is equiangular with the triangle QAB. 43. If from any point in the circumference of a circle a chord and tangent be drawn, the perpendiculars dropped upon them from the middle point of the subtended arc, are equal to one another. IV. 44. 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. Shew that if the distance between the two given points be greater than the sum of their distances from the given line, there will be two such points; if equal, there may be only one; if less, the problem may be impossible. 45. Find the point in a given straight line at which the tangents to a given circle will contain the greatest angle. 46. 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. 47. DF is a straight line touching a circle, and terminated by AD, BF, the tangents at the extremities of the diameter AB, shew that the angle which DF subtends at the center is a right angle. 48. If tangents Am, Bn be drawn at the extremities of the diameter of a semicircle, and any line in mPn crossing them and touching the circle in P, and if AN, BM be joined intersecting in O and cutting the semicircle in E and F; shew that O, P, and the point of intersection of the tangents at E and F, are in the same straight line. 49. If from a point P without a circle, any straight line be drawn cutting the circumference in A and B, shew that the straight lines joining the points A and B with the bisection of the chord of contact of the tangents from P, make equal angles with that chord. V. 50. Describe a circle which shall pass through a given point and which shall touch a given straight line in a given point. 51. Draw a straight line which shall touch a given circle, and make a given angle with a given straight line. 52. Describe a circle the circumference of which shall pass through a given point and touch a given circle in a given point. 53. Describe a circle with a given center, such that the circle so described and a given circle may touch one another internally. 54. Describe the circles which shall pass through a given point and touch two given straight lines. 55. Describe a circle with a given center, cutting a given circle in the extremities of a diameter. 56. 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. 57. 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. VI. 58. If any two circles the centers of which are given, intersect each other, the greatest line which can be drawn through either point of intersection and terminated by the circles, is independent of the diameters of the circles. 59. Two equal circles intersect, the lines joining the points in which any straight line througǹ one of the points of section, which meets the circles with the other point of section, are equal. 60. Draw through one of the points in which any two circles cut one another, a straight line which shall be terminated by their circumferences and bisected in their point of section. 61. Describe two circles with given radii which shall cut each other, and have the line between the points of section equal to a given line. 62. Two circles cut each other, and from the points of intersection straight lines are drawn parallel to one another, the portions intercepted by the circumferences are equal. 63. ACB, ADB are two segments of circles on the same base AB, take any point Cin the segment ACB; join AC, BC, and produce them to meet the segment ADB in D and E respectively: shew that the arc DE is constant. 64. ADB, ACB, are the arcs of two equal circles cutting one another in the straight line AB, draw the chord ACD cutting the inner circumference in C and the outer in D, such that AD and DB together may be double of AC and CB together. 65. If from two fixed points in the circumference of a circle, straight lines be drawn intercepting a given arc and meeting without the circle, the locus of their intersections is a circle. 66. If two circles intersect, the common chord produced bisects the common tangent. 67. Shew that, if two circles cut each other, and from any point in the straight line produced, which joins their intersections, two tangents be drawn, one to each circle, they shall be equal to one another. 68. Two circles intersect in the points A and B; through A and B any two straight lines CAD, EBF, are drawn cutting the circles in the points C, D, E, F; prove that CE is parallel to DF. 69. Two equal circles are drawn intersecting in the points A and B, a third circle is drawn with center A and any radius not greater than AB'intersecting the former circles in D and C. Shew that the three points, B, C, D lie in one and the same straight line. 70. If two circles cut each other, the straight line joining their centers will bisect their common chord at right angles. 71. Two circles cut one another; if through a point of intersection a straight line is drawn bisecting the angle between the diameters at that point, this line cuts off similar segments in the two circles. 72. ACB, APB are two equal circles, the center of APB being on the circumference of ACB, AB being the common chord, if any chord AC of ACB be produced to cut APB in P, the triangle PBC is equilateral. VII. 73. If two circles touch each other externally, and two parallel lines be drawn, so touching the circles in points A and B respectively that neither circle is cut, then a straight line AB will pass through the point of contact of the circles. 74. A common tangent is drawn to two circles which touch each other externally; if a circle be described on that part of it which lies between the points of contact, as diameter, this circle will pass through the point of contact of the two circles, and will touch the line which joins their centers. 75. If two circles touch each other externally or internally, and parallel diameters be drawn, the straight line joining the extremities of these diameters will pass through the point of contact. 76. If two circles touch each other internally, and any circle be described touching both, prove that the sum of the distances of its center from the centers of the two given circles will be invariable. 77. If two circles touch each other, any straight line passing through the point of contact, cuts off similar parts of their circumfe rences. 78. Two circles touch each other externally, the diameter of one being double of the diameter of the other; through the point of contact any line is drawn to meet the circumferences of both; shew that the part of the line which lies in the larger circle is double of that in the smaller. 79. If a circle roll within another of twice its size, any point in its circumference will trace out a diameter of the first. 80. With a given radius, to describe a circle touching two give circles. |