PROBLEMS. BOOK I. The more difficult Problems are distinguished by a space being left at the beginning.] 1. On a given line describe an isosceles triangle, each of whose equal sides shall be double of the base. 2. If two circles cut each other, the line joining their points of intersection is bisected at right angles by the line joining their centres. 3. From every point of a given line, the lines drawn to each of two given points on opposite sides of the line are equal: prove that the line joining the given points will bisect the given line at right angles. 4. The perpendicular is the shortest line that can be drawn from a given point to a given line, and that which is nearer to the perpendicular is less than one more remote; and from the same given point there can be drawn only two equal lines to the given line, one on each side of the shortest line. 5. From a given point without a given line draw a line making a given angle with it. 6. Construct a triangle, having given two sides and an angle opposite to one of them; and shew that there may, according to circumstances, be two solutions, one, or none. 7. Describe a circle which shall pass through two given points, and have its centre in a given line. 8. From two given points on the same side of a given line, draw two lines which shall meet in that line and make equal angles with it. 9. On a given line describe a square of which the line shall be the diagonal. 10. Take any point D in AB one of the sides of a triangle ABC, right-angled at A; in DC take DE equal to AC, and bisect CE in F: then shew that DF, BF, (drawn to a point F within the triangle ABC but not from the extremities of the base) are together greater than AC, BC. 11. The difference of any two sides of a triangle is less than the third side. 12. Let AB bisect CD at right angles in B; from any point E draw EC through the nearest extremity of CD cutting AB in F; and prove that the difference of EF and DF is greater than that of any other two lines drawn from E and D to meet in the line AB. 13. The sum of the diagonals of a quadrilateral is less than the sum of any four lines that can be drawn from any point whatever (except the intersection of the diagonals) to the four angles. 14. Construct a triangle, having given one side, an adjacent angle, and the sum or difference of the other two sides. 15. The centre of a circle being given, find two opposite points in the circumference by means of a pair of compasses only. 16. From a given point draw three lines of given lengths, so that their extremities may be in one line, and equally distant from each other. 17. In the figure, Euc. 1. 5, if BG, CF meet in H, shew that AH bisects the angle BAC. 18. If a line which bisects the vertical angle of a triangle also bisects the base, the triangle is isosceles. 19. From a given point draw a line making equal angles with two given lines. F 20. Through a given point draw a line, so that the perpendiculars upon it from two other given points, may be equal to each other. 21. The lines bisecting the angles of a triangle meet all in one point. 22. The lines bisecting at right angles the sides of a triangle meet all in one point. 23. If two sides of a triangle be produced, the lines, which bisect the two exterior angles and the third interior angle, meet all in one point. 24. If any line, joining two parallel lines, be bisected, any other line, drawn through the point of bisection to meet the two lines, will be bisected in that point. 25. The diameters of a parallelogram bisect each other. 26. A rhombus is a parallelogram, and its diagonals bisect each other at right angles. 27. The line, which joins the middle points of the oblique sides of a trapezium, is equal to the semi-sum of the two parallel sides. 28. The parallelogram, whose diameters are equal, is rectangular. 29. From a given isosceles triangle cut off a trapezium, which shall have the same base as the triangle and its remaining three sides equal to each other. 30. When the corner of a leaf of a book is twice turned down, so that the creases are parallel and the triangular fold of the same breadth as the other, shew that the space included in the second fold is three times that in the first. 31. The quadrilateral figure, whose diameters bisect each other, is a parallelogram. 32. Draw a line which would, if produced, bisect the angle between two given lines, without producing them to meet. 33. Draw a line DE, parallel to the base BC of a triangle ABC, so that DE is equal to the sum of BD and CE. 34. Draw a line DE, parallel to the base BC of a triangle ABC, so that DE is equal to the difference of BD and CE. 35. If AB be bisected in C, and from A, B, C parallel lines be drawn cutting a given line in D, E, F, shew that, according as A and B lie on the same or opposite sides of the line, CF is equal to the semi-sum or semidifference of AD and BE. 36. Through a given point draw a line, such that the parts intercepted by it and two given lines may be equal. 37. If from the base to the sides of an isosceles triangle, three lines be drawn, making equal angles with the base, viz. one from its extremity, the other two from any point in it, these two are together equal to the first. 38. ABCD is a parallelogram; through A draw any line, and shew that the distance of C from this line is equal to the sum or difference of the distances of B and D, according as the line passes without or within the parallelogram. 39. From a given point without the angle contained by two given lines, draw a line so that the part of it between the given point and the nearest line may be equal to the part between the two given lines. 40. Of all triangles, having the same vertical angle, and whose bases pass through a given point, the least is that whose base is bisected in that point. 41. Find a point in the diagonal produced of a square, from which if a line be drawn parallel to any side of the square, and meeting another side produced, it will form, with the produced diagonal and produced side, a triangle equal to the given square. 42. In the figure, Euc. 1. 5, if BG, CF meet in H, and the angles FBG, ABC be equal, then the angle BHF is double of the angle ABC. 43. If in the figure, Euc. 1. 1, CA, CB be produced to meet the circumference in D, E, and F be the other point of intersection of the circles, shew that DF, EF are in one line. 44. Trisect a right angle. 45. One of the acute angles of a right-angled triangle is three times as great as the other: trisect the smaller of these. 46. If the base of an isosceles triangle be produced, twice the exterior angle is greater than two right angles by the vertical angle. 47. If there be an isosceles and an equilateral triangle upon the same base, and so that the inner vertex is equally distant from the other and from the extremities of the base, then, according as that of the former is the inner or outer vertex, its base angle will be 1 or 2 times the vertical. 48. Trisect a given line: and hence divide an equilateral triangle into nine equal parts. 49. The difference of the angles at the base of any triangle is double of the angle between two lines drawn from the vertex, one bisecting the vertical angle and the other perpendicular to the base. 50. In the base BC of an isosceles triangle take any point D; in CA make CE equal to CD, and let EĎ cut AB in F: then shew that three times the angle AEF is greater than four right angles by the angle AFE. 51. If the sides of any polygon be produced to meet, the angles formed by these lines, together with eight right angles, are together equal to twice as many right angles as the figure has sides. 52. If the base angle of an isosceles triangle be onefourth of the vertical angle, and from it a line be drawn perpendicular to the base to meet the opposite side produced, then the part produced, the perpendicular, and the remaining side, will form an equilateral triangle. |