ANALYSIS.-Let AB be the given line, and M and N be the given points without it. Suppose C to be the point of the line AB which is equally distant from M and N. M N B Draw MC and NC, and join M and N. Then since MC = NC, MCN is an isosceles triangle, and a line CD drawn to the middle point of MN will bisect MN at right angles (B. I., Th. 21). From this we derive the following construction; CONSTRUCTION.-Join M and N with the straight line MN; at D, the middle point of MN, erect a to MN. The point C where this intersects the line AB will be the point required. 2. To draw a tangent to a given circle parallel to a given straight line. 3. To draw a tangent to a given circle perpendicular to a given line. 4. To draw a line through a given point so that it will form with the sides of a given angle an isosceles triangle. 5. Through a given point, to draw a straight line which shall make equal angles with two given straight lines. 6. Through a given point P within a given angle, to draw a straight line meeting the sides of the angle and bisected at P. 7. Find a segment of a circle which contains a given acute angle; a given obtuse angle. 8. In a given circle to inscribe a triangle, having given the angles of the triangle. 9. In a given triangle ABC, to draw DE parallel to the base AB, so that DEAD + BE. 10. From two given points on the same side of a given straight line, to draw lines meeting the given line and making equal angles with it. 11. To draw a tangent common to two given circles. CONSTRUCTION OF POLYGONS. To construct a square, having given : 12. Its perimeter. 13. Its diagonal. 14. Its inscribed circle. 15. An inscribed triangle. To construct an equilateral triangle, having given: 16. The perimeter. 17. The altitude. To construct an isosceles triangle, 20. The base and altitude. 18. The inscribed circle. 22. The perimeter and the altitude. 24. The base and inscribed circle. 25. The base and circumscribed circle. To construct a right triangle, having given 26. The hypotenuse and either leg. 27. The hypotenuse and either acute angle. 28. The altitude upon the hypotenuse and the hypotenuse. 29. The altitude upon the hypotenuse and one side. 30. The radius of the inscribed circle and one leg. 31. The radius of the inscribed circle and hypotenuse. 32. The radius of the inscribed circle and an acute angle. 33. The sum, or the difference, of the legs and an acute angle. To construct a triangle, having given: 34. The base and two angles. 35. The base, altitude, and an angle at the base. 36. The base, altitude, and vertical angle. 37. The base, median to the base, and vertical angle. 38. The base, an angle at the base, and the sum of the other sides. 39. The base, an angle at the base, and the difference of the other sides. 40. The base, vertical angle, and sum of the other sides. 41. The base, vertical angle, and difference of the other sides. 42. Two angles and the sum of two sides. 43. Two angles and the sum of the three sides. 44. The three angles and the radius of the inscribed circle. 45. The three angles and the radius of the circumscribed circle. 46. An angle, its bisector, and altitude let fall from the angle. 47. The middle points of its sides. Its three medians. To construct a rectangle, having given : -48. One side and its diagonal. 49. One side and the angle of a diagonal. 50. The perimeter and the diagonal. 51. The sum of two adjacent sides and the angle of a diagonal. 52. The difference of two adjacent sides and the angle of a diagonal. To construct a rhombus, having given: 53. The two diagonals. 54. One angle and one diagonal. 55. Two adjacent sides and the included angle. 56. Two adjacent sides and the diagonal opposite them. 57. One side and the radius of the inscribed circle. 58. One angle and the radius of the inscribed circle. To construct a rhomboid, having given: 59. One side and the two diagonals. 60. One side, one angle, and one diagonal. 61. The two diagonals and their included angle. 62. The base, the altitude, and either angle. To construct an isosceles trapezoid, having given: 63. The bases and the altitude. 64. The bases and the diagonal. 65. The bases and one angle. 66. The bases and radius of the circumscribed circle. To construct a trapezoid, having given: 67. The two bases and the two legs. 68. The two bases and the two diagonals. 69. The two bases, one leg, and its angle with either base. 70. The two bases, one diagonal, and the angle included between the diagonals. CONSTRUCTION OF CIRCLES. To construct a circle, with a given radius, which shall: 71. Touch each of two intersecting lines. 72. Touch a given line and a given circle. 73. Pass through a given point and touch a given line. 74. Pass through a given point and touch a given circle. To construct a circle which shall be tangent: 75. To three given lines, two of which are parallel. 76. To two given parallels, and pass through a given point. 77. To a given line at a given point, and pass through another given point. 78. To two given lines, and at a given point of one of them. 79. To a given circle at a given point, and pass through another given point. 80. To a given line, and also to a given circle at a given point. 81. To a given circle, and also to a given line at a given point. 82. To divide a circle into two segments such that the angle contained in one part shall be twice the angle contained in the other part. SUGGESTION.-Inscribe in the circle an equilateral triangle. 83. To divide a circle into two segments such that the angle contained in one part shall be five times as great as the angle contained in the other part. SUGGESTION.-Inscribe in the circle a regular hexagon. THEOREMS IN BOOK IV. 1. THE area of an equilateral triangle is equal to a side into threefourths of the radius of the circumscribed circle. 2. The area of a circumscribed polygon is equal to half the product of the perimeter by the radius of the circle inscribed. 3. The area of a rhombus is equal to one-half of the product of its diagonals. 4. The perimeter of a triangle is to one of its sides as the altitude let fall on that side is to the radius of the inscribed circle. 5. The sum of the perpendiculars from any point within a convex equilateral polygon upon the sides is constant. 6. Two quadrilaterals are equivalent if two of their diagonals are respectively equal and contain equal angles with the sides. 7. In an isosceles right triangle either leg is a mean proportional between the hypotenuse and the perpendicular upon the hypotenuse from the vertex of the right angle. 8. Two triangles are equivalent if they have two sides of the one respectively equal to two sides of the other and the included angles are supplements of each other. 9. Two parallelograms are equivalent if two adjacent sides of one are respectively equal to two adjacent sides of the other, and the included angles are supplementary. 10. The two opposite triangles formed by joining any point within a parallelogram to its four vertices are together equivalent to one half the parallelogram. 11. If the middle point of one of the non-parallel sides of a trapezoid is joined with the extremities of the opposite side, the triangle formed is one half the trapezoid. 12. If lines are drawn from two opposite vertices of a parallelogram to the middle points of the sides, the parallelogram formed will be one half of the given parallelogram. 13. The square on the hypotenuse of an isosceles right triangle is equal to four times the square on the perpendicular drawn from the right angle to the hypotenuse. 14. If a line is drawn through the point of intersection of the diagonals of a parallelogram, it divides the parallelogram into two equal quadrilaterals. 15. If lines are drawn from the middle point of the diagonal of a quadrilateral to opposite vertices, they will divide the quadrilateral into two equivalent figures. 16. If from any point in the diagonal of a parallelogram, straight lines are drawn to the opposite angles, they will cut off equivalent triangles. 17. If two triangles have an angle in common and their areas are equal, the sides about the equal angles are reciprocally proportional. 18. The diagonals of a trapezoid divide each other into segments which are proportional. 19. The perpendicular from any point of a circumference upon a chord is a mean proportional between the perpendiculars from the same point upon the tangents drawn at the extremities of the chord. 20. In any triangle ABC, if from the vertex C CE is drawn to the circumference of the circumscribed circle, and CD to the base BC, making the angle ACD equal to BCE, then ACX BC=CD×CE. 21. In any triangle ABC, if from the vertex C, CE is drawn bisecting the angle C and meeting the circumference of the circumscribed circle in E and the base AB in D, then ACX BC= CDX CE. E B 22. The sum of the squares of the four sides of any quadrilateral is equal to the sum of the squares of the diagonals, increased by four times the square of the line joining the middle points of the diagonals. PROBLEMS IN BOOK IV. 1. To construct a triangle, having given its angles and its area. SUGGESTION.-Find a square equal to the area. Construct any triangle having the given angles; then see B. IV., Prob. 19. 2. To construct a triangle equivalent to a given triangle and having one side equal to a given length. 3. To construct a triangle, having given one angle, the side opposite, and the area. 4. To construct a right triangle equivalent to a given triangle. 5. To construct an isosceles triangle equivalent to a given triangle. 6. To construct an equilateral triangle equivalent to a given triangle. 7. Given any triangle, to construct an equivalent triangle having one side equal to a given length and one angle equal to an angle of the given triangle. 8. Given any triangle, to construct an equivalent right triangle having one leg equal to a given length. 9. Given any triangle, to construct an equivalent right triangle having the hypotenuse equal to a given length. |