EXERCISES ON BOOK I. 1. If a perpendicular be drawn from the iniddle point of a given straight line, any point in it is equally distant from the extremities of the given line. 2. If the opposite sides of a quadrilateral be equal, and its angles right angles, its diagonals will be equal. 3. The straight lines bisecting the base angles of an isosceles triangle, form with the base an isosceles triangle. 4. If the two adjacent sides BA and AD of a quadrilateral ABCD be equal, and if the diagonal AC bisect the angle A, then shall the sides BC and CD be equal, and also the angles which they make with the diagonal. 5. The lines which bisect the angles at the base of an isosceles triangle, and meet the opposite sides, are equal. 6. The straight lines drawn from the angles of an equilateral triangle to the points of bisection of the opposite sides are equal. 7. The opposite angles of a rhombus are equal; the diagonals bisect the angles, and bisect one another at right angles. 8. If two isosceles triangles be described upon a given line, the straight line which joins their vertices, or that line produced, bisects the given line at right angles. 9. In an isosceles triangle (1) The line which bisects the vertical angle also bisects the base. (2) The line which joins the middle of the base with the vertex, bisects the vertical angle, and is perpendicular to the base. (3) The perpendiculars drawn from the extremities of the base to the opposite sides are equal. 10. In a triangle, if the perpendicular from the vertex on the base bisect the base, the triangle is isosceles. 11. In the figure of I. 5, let O be the point of intersection of BG and CF, and shew that FO and GO are equal. 12. If the point O, as in Ex. 11, be joined to A, shew that OA bisects the angle A. 13. If any angle and its supplement be bisected, the bisecting lines are at right angles to one another. 14. If three points be taken on the sides of an equilateral triangle, at equal distances from the angles, and the points joined, the triangle so formed will also be equilateral. 15. Prove that any two sides of a triangle are together greater than twice the line drawn from the angle between them to the middle of the opposite side. 16. The sides of a four-sided figure are together greater than the sum of the diagonals. 17. The two sides of a triangle are together greater than double the straight line bisecting the vertical angle, and meeting the base. 18. 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. 19. In an equilateral triangle, the perpendiculars drawn from the angles to the opposite sides are equal, and the three perpendiculars divide the triangle into six equal triangles. 20. If a straight line bisect the base and vertical angle of a triangle. the triangle is isosceles. 21. In the figure of I. 16, shew by how much the inner vertical angle O exceeds the outer vertical angle A. 22. If from any point in the base of an equilateral triangle parallels to the sides be drawn, a parallelogram is formed whose perimeter is constant. 23. The middle point of the hypotenuse of a right-angled triangle is equally distant from the three angles. 24. If four points be taken in the sides of a square, at equal distances from the angles, the figure formed by joining them will also be a square. 25. To bisect a triangle by a straight line drawn from a given point in one of its sides. 26. To bisect a parallelogram by a straight line drawn from a given point in one of its sides. 27. To bisect a trapezium by a straight line drawn from one of its angles. 28. To trisect a given finite straight line. 29. To describe an isosceles right-angled triangle on a given finite straight line. 30. Given the diagonal of a square to construct the square. 31. 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 at the base of the triangle. 32. In a given right-angled isosceles triangle to inscribe a square. 33. To inscribe a square in a given equilateral triangle. 34. Given the perpendicular of an equilateral triangle to construct the triangle. 35. Shew what are the magnitudes with respect to a right angle of the angles of the following regular polygons: The pentagon, hexagon, heptagon, octagon, nonagon, and decagon. 36. Shew that three regular hexagons can be placed so as to have a common point, and to fill up the space round that point. 37. Shew that two regular octagons and one square have the same property. Draw patterns illustrating this and the preceding. 38. Find a point equally distant from two given points. How many points are there which fulfil this condition? 39. Find a point in a given line equally distant from two given points. How many points fulfil the conditions? When is the problem impossible? 40. Find a point in the circumference of a given circle equally distant from two given points. How many points fulfil the conditions? When is the problem impossible? 41. Find a point equally distant from three given points. When is the problem impossible? 42. Find a point equally distant from a given point, and from a given line. BOOK II. PROPOSITION I.-THEOREM. (Etc. II. 1.) If there be two straight lines, one of which is divided into any number of parts, the rectangle contained by the two straight lines is equal to the sum of the rectangles contained by the undivided line, and the several parts of the divided line. Let AB and C be the two straight lines, of which AB is divided into any numbers of parts at the points D, E. It is required to prove that the rectangle contained by AB and C, is equal to the several rectangles contained by AD and C, DE and C, EB and C. D E B From A draw AM perpendicular to AB (I. 7, cor.), and cut off AF equal to C. Through F draw FL parallel to AB (I. 22), meeting in K the line through B parallel to AM. Through D and E draw F DG and EH parallel to AF (I. 22). Then the figures so formed, and M also AK, are all rect angles, so that DG and EH are each equal to AF, that is to C (I. 24). Now the whole rectangle AK is equal to the sum of the rectangles AG, DH, and EK. But AK is the rectangle contained by AB and C, (Ax. 10.) for it is contained by AB and AF, of which AF is equal to C; also AG is the rectangle contained by AD and C, for AF is equal to C; DH is the rectangle contained by DE and C; and EK the rectangle contained by EB and C. Therefore the rectangle contained by AB and C is equal to the several rectangles contained by AD and C, DE and C, EB and C. Q. E. D. 43. Find a point equally distant from two given lines. How many points fulfil this condition? 44. In a given line find a point such that the lines drawn from it to two given points make equal angles with the given line (1) When the points are on the same side of the line. (2) When they are on opposite sides of it. 45. From a given point, without a given line, to draw a line making a given angle with the given line. 46. Construct an isosceles triangle, having given— (1) The base and altitude. (2) The altitude and an angle at the base. (3) The altitude and the vertical angle. (4) The base and vertical angle. (5) The altitude and the sum of the equal sides. (6) The altitude and the difference between it and one of the equal sides. 47. Construct a right-angled triangle, having given (1) The hypotenuse and one of the acute angles. (2) The hypotenuse and one other side. (3) The hypotenuse and the sum of the other two sides. (4) The hypotenuse and the difference of the other two sides. 48. Construct a triangle, having given (1) The base, and each of the angles at the base. (2) The base, the vertical angle, and one of the angles at the base. (3) The base, one of the angles at the base, and the sum of the other two sides. (4) The base, one of the angles at the base, and the difference of the other two sides. (5) The base, the difference of the angles at the base, and the difference of the other two sides. (6) The base, the sum of the other two sides, and also their difference. (7) The base, the altitude, and also one of the angles at the base. (8) The middle points of its three sides. (9) Two sides and the line from the vertical angle to the middle of the base. 49. If from the diagonal BD of a square ABCD, BE is cut off equal to BC, and EF is drawn perpendicular to BD, meeting CD in F, prove CF, FE, ED all equal to one another. Hence construct a square, having given the difference between the diagonal and the side. 50. 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. |