EXERCISES IN EUCLID. I. 1 to 15. 1. On a given straight line describe an isosceles triangle having each of the sides equal to a given straight line. 2. In the figure of I. 2 if the diameter of the smaller circle is the radius of the larger, shew where the given point and the vertex of the constructed triangle will be situated. 3. If two straight lines bisect each other at right angles, any point in either of them is equidistant from the extremities of the other. 4. If the angles ABC and ACB at the base of an isosceles triangle be bisected by the straight lines BD, CD, shew that DBC will be an isosceles triangle. 5. BAC is a triangle having the angle B double of the angle A. If BD bisects the angle B and meets AC at D, shew that BD is equal to AD. 6. In the figure of I. 5 if FC and BG meet at H shew that FH and GH are equal. 7. In the figure of I. 5 if FC and BG meet at H, shew that AH bisects the angle BAC. 8. The sides AB, AD of a quadrilateral ABCD are equal, and the diagonal AC bisects the angle BAD: shew that the sides CB and CD are equal, and that the diagonal AC bisects the angle BCD. 9. ACB, ADB are two triangles on the same side of AB, such that AC is equal to BD, and AD is equal to BC, and AD and BC intersect at 0: shew that the triangle AOB is isosceles. 10. The opposite angles of a rhombus are equal. 11. A diagonal of a rhombus bisects each of the angles through which it passes. 12. If two isosceles triangles are on the same base the straight line joining their vertices, or that straight line produced, will bisect the base at right angles. 13. Find a point in a given straight line such that its distances from two given points may be equal. 14. Through two given points on opposite sides of a given straight line draw two straight lines which shall meet in that given straight line, and include an angle bisected by that given straight line. 15. A given angle BAC is bisected; if CA is produced to G and the angle BAG bisected, the two bisecting lines are at right angles. 16. If four straight lines meet at a point so that the opposite angles are equal, these straight lines are two and two in the same straight line. I. 16 to 26. 17. ABC is a triangle and the angle A is bisected by a straight line which meets BC at D; shew that BA is greater than BD, and CA greater than CD. 18. In the figure of I. 17 shew that ABC and ACB are together less than two right angles, by joining A to any point in BC. 19. ABCD is a quadrilateral of which AD is the longest side and BC the shortest; shew that the angle ABC is greater than the angle ADC, and the angle BŬD greater than the angle BAD. 20. If a straight line be drawn through A one of the angular points of a square, cutting one of the opposite sides, and meeting the other produced at F, shew that AF is greater than the diagonal of the square. 21. The perpendicular is the shortest straight line that can be drawn from a given point to a given straight line; and of others, that which is nearer to the perpendicular is less than the more remote; and two, and only two, equal straight lines can be drawn from the given point to the given straight line, one on each side of the perpendicular. 22. The sum of the distances of any point from the three angles of a triangle is greater than half the sum of the sides of the triangle. 23. The four sides of any quadrilateral are together greater than the two diagonals together. 24. The two sides of a triangle are together greater than twice the straight line drawn from the vertex to the middle point of the base. 25. If one angle of a triangle is equal to the sum of the other two, the triangle can be divided into two isosceles triangles. 26. If the angle Cof a triangle is equal to the sum of the angles A and B, the side AB is equal to twice the straight line joining C to the middle point of AB. 27. Construct a triangle, having given the base, one of the angles at the base, and the sum of the sides. 28. The perpendiculars let fall on two sides of a triangle from any point in the straight line bisecting the angle between them are equal to each other. 29. In a given straight line find a point such that the perpendiculars drawn from it to two given straight lines shall be equal. 30. Through a given point draw a straight line such that the perpendiculars on it from two given points may be on opposite sides of it and equal to each other. 31. A straight line bisects the angle A of a triangle ABC; from B a perpendicular is drawn to this bisecting straight line, meeting it at D, and BD is produced to meet AC or AC produced at E: shew that BD is equal to DE. 32. AB, AC are any two straight lines meeting at A: through any point P draw a straight line meeting them at E and F, such that AE may be equal to AF. 33. Two right-angled triangles have their hypotenuses equal, and a side of one equal to a side of the other: shew that they are equal in all respects. I. 27 to 31. 34. Any straight line parallel to the base of an isosceles triangle makes equal angles with the sides. 35. If two straight lines A and B are respectively parallel to two others C and D, shew that the inclination of A to B is equal to that of C to D. 36. A straight line is drawn terminated by two parallel straight lines; through its middle point any straight line is drawn and terminated by the parallel straight lines. Shew that the second straight line is bisected at the middle point of the first. 37. If through any point equidistant from two parallel straight lines, two straight lines be drawn cutting the parallel straight lines, they will intercept equal portions of these parallel straight lines. 38. If the straight line bisecting the exterior angle of a triangle be parallel to the base, shew that the triangle is isosceles. 39. Find a point B in a given straight line CD, such that if AB be drawn to B from a given point A, the angle ABC will be equal to a given angle. 40. If a straight line be drawn bisecting one of the angles of a triangle to meet the opposite side, the straight lines drawn from the point of section parallel to the other sides, and terminated by these sides, will be equal. 41. The side BC of a triangle ABC is produced to a point D; the angle ACB is bisected by the straight line CE which meets AB at E. A straight line is drawn through E parallel to BC, meeting AC at F, and the straight line bisecting the exterior angle ACD at G. Shew that EF is equal to FG. 42. AB is the hypotenuse of a right-angled triangle ABC: find a point D in AB such that DB may be equal to the perpendicular from D on AC. 43. ABC is an isosceles triangle: find points D, E in the equal sides AB, AC such that BD, DE, EC may all be equal. 44. A straight line drawn at right angles to BC the base of an isosceles triangle ABC cuts the side AB at D and CA produced at E: shew that AED is an isosceles triangle. I. 32. 45. From the extremities of the base of an isosceles triangle straight lines are drawn perpendicular to the sides; shew that the angles made by them with the base are each equal to half the vertical angle. 46. On the sides of any triangle ABC equilateral triangles BCD, CAE, ABF are described, all external: shew that the straight lines AD, BE, CF are all equal. 47. What is the magnitude of an angle of a regular octagon ? 48. Through two given points draw two straight lines forming with a straight line given in position an equilateral triangle. 49. If the straight lines bisecting the angles at the base of an isosceles triangle be produced to meet, they will contain an angle equal to an exterior angle of the triangle. 50. A is the vertex of an isosceles triangle ABC, and BA is produced to D, so that AD is equal to BA; and DC is drawn: shew that BCD is a right angle. 51. ABC is a triangle, and the exterior angles at B and Care bisected by the straight lines BD, CD respectively, meeting at D: shew that the angle BDC together with half the angle BAC make up a right angle. 52. Shew that any angle of a triangle is obtuse, right, or acute, according as it is greater than, equal to, or less than the other two angles of the triangle taken together. 53. Construct an isosceles triangle having the vertical angle four times each of the angles at the base. 54. In the triangle ABC the side BC is bisected at E and AB at G; AE is produced to F so that EF is equal to AE, and CG is produced to H so that GH is equal to CG: shew that FB and HB are in one straight line. 55. Construct an isosceles triangle which shall have one-third of each angle at the base equal to half the vertical angle. 56. AB, AC are two straight lines given in position: it is required to find in them two points P and Q, such that, PQ being joined, AP and PQ may together be equal to a given straight line, and may contain an angle equal to a given angle. 57. Straight lines are drawn through the extremities of the base of an isosceles triangle, making angles with it on the side remote from the vertex, each equal to one-third of one of the equal angles of the triangle and meeting the sides produced: shew that three of the triangles thus formed are isosceles. 58. AEB, CED are two straight lines intersecting at E; straight lines AC, DB are drawn forming two triangles ACE, BED; the angles ACE, DBE are bisected by the straight lines CF, BF, meeting at F. Shew that the angle CFB is equal to half the sum of the angles EAC, EDB. |