BOOK XIII. MISCELLANEOUS GEOMETRICAL EXERCISES. 1. IF the opposite angles formed by four lines meeting at a point are equal, these lines form but two straight lines. 2. If the equal sides of an isosceles triangle are produced, the two exterior angles formed with the base will be equal. 3. The sum of any two sides of a triangle is greater than the third side. 4. If from any point within a triangle two straight lines. are drawn to the extremities of either side, they will include a greater angle than that contained by the other two sides. 5. If two quadrilaterals have the four sides of the one equal to the four sides of the other, each to each, and the angle included by any two sides of the one equal to the angle contained by the corresponding sides of the other, the quadrilaterals are themselves equal. 6. The sum of the diagonals of a trapezium is less than the sum of any four lines which can be drawn to the four angles from any point within the figure, except from the intersection of the diagonals. 7. Lines joining the corresponding extremities of two equal and parallel straight lines, are themselves equal and parallel, and the figure formed is a parallelogram. 8. If, in the sides of a square, at equal distances from the four angles, points be taken, one in each side, the straight lines joining these points will form a square. 9. If one angle of a parallelogram is a right angle, all its angles are right angles. 10. Any straight line drawn through the middle point of a diagonal of a parallelogram to meet the sides, is bisected in that point, and likewise bisects the parallelogram. 11. If four magnitudes are proportionals, the first and second may be multiplied or divided by the same magnitude, and also the third and fourth by the same magnitude, and the resulting magnitudes will be proportional. 12. If four magnitudes are proportionals, the first and third may be multiplied or divided by the same magnitude, and also the second and fourth by the same magnitude, and the resulting magnitudes will be proportionals. 13. If there be two sets of proportional magnitudes, the quotients of the corresponding terms will be proportionals. 14. If any two points be taken in the circumference of a circle, the straight line joining them will lie wholly within the circle. 15. The diameter is the longest straight line that can be inscribed in a circle. 16. If two straight lines intercept equal arcs of a circle, and do not cut each other within the circle, the lines will be parallel. 17. If a straight line be drawn to touch a circle, and be parallel to a chord, the point of contact will be the middle point of the arc cut off by that chord. 18. If two circles cut each other, and from either point of intersection diameters be drawn, the extremities of these diameters and the other point of intersection will be in the same straight line. 19. If one of the equal sides of an isosceles triangle be the diameter of a circle, the circumference of the circle will bisect the base of the triangle. 20. If the opposite angles of a quadrilateral be together equal to two right angles, a circle may be circumscribed about the quadrilateral. 21. Parallelograms which have two sides and the included angle equal in each, are themselves equal. 22. Equivalent triangles upon the same base, and upon the same side of it, are between the same parallels. 23. If the middle points of the sides of a trapezoid, which are not parallel, be joined by a straight line, that line will be parallel to each of the two parallel sides, and be equal to half their sum. 24. If, in opposite sides of a parallelogram, at equal distances from opposite angles, points be taken, one in each side, the straight line joining these points will bisect the parallelogram. 25. The perimeter of an isosceles triangle is greater than the perimeter of a rectangular parallelogram, which is of the same altitude with, and equivalent to, the given triangle. 26. If the sides of the square described upon the hypothenuse of a. right-angled triangle be produced to meet the sides (produced if necessary) of the squares described upon the other two sides of the triangle, they will form. triangles equiangular with and equal to the given triangle. 27. A square circumscribed about a given circle is double a square inscribed in the same circle. 28. If the sum of the squares of the four sides of a quadrilateral be equivalent to the sum of the squares of the two diagonals, the figure is a parallelogram. 29. Straight lines drawn from the vertices of a triangle, so as to bisect the opposite sides, bisect also the triangle. 30. The straight lines which bisect the three angles of a triangle meet in the same point. 31. The area of a triangle is equal to its perimeter multiplied by half the radius of the inscribed circle. 32. If the points of bisection of the sides of a given triangle be joined, the triangle so formed will be one fourth. of the given triangle. 33. To describe a square upon a given straight line. 34. To find in a given straight line a point equally distant from two given points. 35. To construct a triangle, the base, one of the angles. at the base, and the sum of the other two sides being given. 36. To trisect a right angle. 37. To divide a triangle into two parts by a line drawn parallel to a side, so that these parts shall be to each other as two given straight lines. 38. To divide a triangle into two parts by a line drawn perpendicular to the base, so that these parts shall be to each other as two given lines. 39. To divide a triangle into two parts by a line drawn from a given point in one of the sides, so that the parts shall be to each other as two given lines. 40. To divide a triangle into a square number of equal triangles, similar to each other and to the original triangle. 41. To trisect a given straight line. 42. To inscribe a square in a given right-angled isosceles triangle. 43. To inscribe a square in a given quadrant. 44. To describe a circle that shall pass through a given point, have a given radius, and touch a given straight line. 45. To describe a circle, the centre of which shall be in the perpendicular of a given right-angled triangle, and the circumference of which shall pass through the right angle and touch the hypothenuse. 46. To describe three circles of equal diameters which shall touch each other, and to describe another circle which shall touch the three circles. 47. If, on the diameter of a semicircle, two equal circles be described, and in the curvilinear space included by the three circumferences a circle be inscribed, its diameter will be to that of the equal circles in the ratio of two to three. 48. If two points be taken in the diameter of a circle, equidistant from the centre, the sum of the squares of two lines drawn from these points to any point in the circumference will always be the same. 49. Given the vertical angle, and the radii of the inscribed and circumscribed circles, to construct the triangle. 50. If a diagonal cuts off three, five, or any odd number of sides from a regular polygon, the diagonal is parallel to one of the sides. 51. The area of a regular hexagon inscribed in a circle. is double that of an equilateral triangle inscribed in the same circle. 52. The side of a square circumscribed about a circle is equal to the diagonal of a square inscribed in the same circle. 53. To describe a circle equal to half a given circle. 54. A regular duodecagon is equivalent to three fourths of the square constructed on the diameter of its circumscribed circle; or is equal to the square constructed on the side of the equilateral triangle inscribed in the same circle. 55. If semicircles be described on the sides of a rightangled triangle as diameters, the one described on the hypothenuse will be equal to the sum of the other two. 56. If on the sides of a triangle inscribed in a semicircle, semicircles be described, the two crescents thus formed will together equal the area of the triangle. 57. If the diameter of a semicircle be divided into any number of parts, and on them semicircles be described, their circumferences will together be equal to the circumference of the given semicircle. 58. To divide a circle into any number of parts, which shall all be equal in area and equal in perimeter, and not have the parts in the form of sectors. 59. To draw a straight line perpendicular to a plane, from a given point above the plane. 60. Two straight lines not in the same plane being |