16. The line bisecting an inscribed angle bisects the intercepted arc. 17. The line joining the vertex of an inscribed angle to the middle point of its intercepted arc bisects the angle. 18. The sum of the interior angles of a triangle equals two right angles. (Prove by circumscribing a circle and drawing a tangent at one vertex.) 19. The circle drawn on one of the sides of an isosceles triangle as a diameter bisects the base. 20. An inscribed parallelogram is a rectangle. 21. The diagonals of a rectangle inscribed in a circle are diameters. 22. A circumference drawn on the hypotenuse of a right triangle as a diameter passes through the vertices of all right triangles having the same hypotenuse. 23. If two circles are tangent externally at C, and a common external tangent touches them at A and B, the angle ACB is a right angle. (Draw the internal tangent.) 24. A circle drawn on the radius of another circle as a diameter bisects all chords of the larger circle drawn from the point of tangency. 25. If a circle is drawn upon one arm of a right angle triangle as a diameter, and a tangent is drawn at the point of its intersection with the hypotenuse, this tangent bisects the other arm. 26. If two equal circles are tangent and a straight line is drawn through the point of tangency and ending in the circumferences, the parts of the line are equal. 27. What is the locus of all points from which two equal tangents can be drawn to two tangent circles? 28. What is the locus of the vertices of the triangles which have a given acute angle and a given side opposite? 29. Inscribe a circle in a sector. 30. Inscribe a circle in a quadrilateral having two opposite pairs of adjacent sides equal. 31. The line of centers of two equal circles bisects their common internal tangent. 32. If semicircles are drawn on the sides of an equilaterial triangle, these arcs intersect at the middle points of the sides of the triangle. 33. Find the locus of the points of intersection of the diagonals of trapezoids on the same base and with the same base angles. 34. Two circles intersect at C and D. Diameters CA and CB are drawn, and chords AD and BD. Prove ADB a straight line. 35. In an equilateral triangle construct three equal circles each tangent to two sides and to two circles. 36. In a given circle construct three equal circles each tangent to the given circle and to two others. 37. Through a given point on the bisector of an angle construct a circle tangent to the sides of the angle. 38. Construct a circle passing through a given point and tangent to a given line at a given point. 39. Two equal circles are tangent externally. Find the locus of the centers of circles tangent to both given circles externally. 40. By means of a circle construct a right triangle, having given the hypotenuse and an arm. 41. By means of a circle construct a right triangle, having given the hypotenuse and an acute angle. 42. Also, having given the hypotenuse and the altitude upon the hypotenuse. 43. If the angle between two tangents to a circle is 40°, how many degrees in each intercepted arc? 44. Through two given points draw parallel lines at a given distance apart. 45. Construct an isosceles right triangle having given the hypotenuse. 46. Construct a triangle ABC, given AB and the median to AB 214". 47. Construct a right triangle, having given an acute angle and the sum of the arms. = = 2" (inches), B = 75°, 48. Construct an isosceles triangle, having given the perimeter and the altitude. 49. Construct a triangle, having given one side, an adjacent angle, and the sum of the other two sides. Construct an equilateral triangle, having given (a) the perimeter; (b) the altitude; (c) the median; (d) the radius of the inscribed circle; (c) the radius of the circumscribed circle. 51. Construct a square, having given the sum of the diagonal and a side; having given a diagonal. 52. Construct a triangle having given A, B, and the radius of the circumscribed circle. 53. Construct a regular dodecagon, having given a side. 54. Construct a common external tangent to two unequal circles. 55. Construct a common internal tangent to two unequal circles. Construct a triangle having given: 56. a, ha, ma. 57. ha, hb, L C. 58. a, LA, ha. 59. a, b, hb. ma, ha, LB. 61. a, b, LA + Z B. Construct a rectangle having given: 62. A side and a diagonal. 63. A side and the angle formed by the diagonals. 64. The perimeter and a diagonal. 65. A diagonal and the angle formed by the diagonals. Construct a rhombus having given: 66. A side and a diagonal. 67. The perimeter and a diagonal. 68. One angle and a diagonal. 69. The two diagonals. Construct a parallelogram, having given: 70. Two non-parallel sides and an angle. 71. One side and the diagonals. 72. One side, an angle, and the altitude to that side. 73. The diagonals and the angle between them. 74. Two non-parallel sides and the altitude. 75. Two non-parallel sides and a diagonal. A trapezoid can always be divided into a parallelogram and a triangle. Therefore, in constructing a trapezoid, construct the parallelogram and the triangle. Construct a trapezoid having given: 76. The bases and a diagonal. (Isosceles.) 77. Two sides and their included angle. (Isosceles.) 78. The bases and the angles adjacent to one base. 79. The bases, the altitude, and an angle. 80. The bases and the altitude. (Isosceles.) Construct a circle that will: 81. Touch a given circle at P, and pass through a point Q. 82. Pass through point P, and touch two given parallels. 83. Touch two given lines, one of them at a point P. 84. Touch three given non-parallel lines. 85. Touch two given circles and have a given radius. 86. Construct a triangle, having given the mid-points of the three sides. PROPORTION CHAPTER XVI PROPORTION FROM PARALLEL LINES The basic method is: *237. Three or more parallel lines intercept proportional parts on two transversals. 241. Define Alternation, Inversion, Composition, and Division in proportion. 242. The product of the means equals what? 243. What is the converse of Art. 242? 244. If three terms of one proportion are equal respectively to three terms of another proportion, what can be said of the remaining terms? 245. A straight line parallel to one side of a triangle divides the other two sides proportionally. 246. Arrange the proportion of 245 by composition. *247. PROBLEM: Find the fourth proportional to three given straight lines. 248. PROBLEM: Divide a given straight line into parts proportional to any number of given straight lines. *249. The bisector of an angle of a triangle divides the opposite side into parts proportional to the other two sides. 250. Explain Internal division of a line. External divi sion. *251. The bisector of an exterior angle of a triangle divides the opposite side, externally, into parts proportional to the other two sides. *252. A line dividing two sides of a triangle proportionally is parallel to the third side. 253. Define Similar Polygons. SUMMARY 254. State two methods of proving lines in proportion. 255. State a new method of proving lines parallel. 256. EXERCISES 1. If a line is parallel to two sides of a parallelogram and intersects the other two sides, it divides these sides proportionally. 2. If a line is parallel to the bases of a trapezoid and intersects the other two sides, it divides these sides proportionally. 3. Prove Art. 106, ex. 13, by Art. 252; also ex. 14, 15, and 16. 4. Prove Art. 116 by Art. 252. 5. If a line divides two opposite sides of a parallelogram proportionally, it is parallel to the other sides. 6. If a line divides the non-parallel sides of a trapezoid proportionally, it is parallel to the bases. 7. If a, b, and c are the first three terms of a proportion, show bc that the fourth proportional x = a 8. Find the fourth proportional to 2, 3, and 4, arithmetically and geometrically. 9. Find the fourth proportional to 3, 5, and 2, arithmetically and geometrically. = 10. In triangle ABC, AB 6; AC 5; a line EF || BC divides AC into parts; AF = 4 and FC 1. Find AE and EB. 11. In ▲ABC, EF || BC; AF=2EB; AE=20; FC=10. Find AF. 12. In triangle ABC, AB = 12; AC of BC made by the bisector of angle A. 13. In the same figure, find the parts of AC made by the bisector of angle B; also of AB similarly. = 14; BC= 13. Find the parts = = = |