I. 5. Two pairs of equal straight lines being given, shew how to construct with them the greatest parallelogram. 6. Within a square there is no point so distant from the point of intersection of the diagonals as the corners. 7. Determine the shortest path from one given point to another, subject to the condition, that it shall meet two given lines. -8. A point is taken within a square, and straight lines drawn from it to the angular points of the square, and perpendicular to the sides; the squares on the first are double the sum of the squares on the last. Shew that these sums are least when the point is in the center of the square, 9. To find a point in a given right line such that the sum of the squares on the lines from the two points to that point shall be less than that of the lines from the two given points to any other point in the given line, 10. The sum of the squares of two lines is never less than twice their rectangle. 11. Divide a given straight line so that the rectangle under the parts may be equal to a given square, and point out the limit which the side of the given square must not exceed so that the problem may be possible. 12. If two straight lines of given magnitude cut each other at right angles, the sum of the rectangles contained by the segments into which each divides the other is least when they bisect each other, II. 13. The perimeter of a square is less than that of any other parallelogram of equal area. 14. Shew that of all quadrilateral figures having the same perimeter, that which is a square is the greatest. 15. Show that of all equiangular parallelograms of equal perimeters, that what is equilateral is the greatest. 16. Prove that the perimeter of an isosceles triangle is greater than that of an equal right-angled parallelogram of the same altitude. 17. 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 their intersection. 18. If two sides of a triangle be given, the triangle will be greatest when they contain a right angle. 19. Given one of the angles and the perimeter of a plane triangle, to find the sides, when the area is the greatest possible. 20. ABC is a right-angled triangle; find the point P in AC, so that the sum of the distances from A and AC is the least possible. 21. Given the base and one side of a triangle, to find the third side, so that the area may be the greatest possible. 22. If from a point in the base of an isosceles triangle, two straight lines be drawn parallel to and terminated by the sides, the sum of the areas of the new triangles so formed will be least when the point is at the center of the base. 23. 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 the given point. 24. Of all the triangles having the same base and the same perimeter, that is the greatest which has the two undetermined sides equal. 25. The perimeter of an isosceles triangle is less than that of any other equal triangle upon the same base. 26. Of all triangles on the same base, having equal perimeters, the equilateral has the greatest area. 27. Find the greatest of all triangles having the same vertical angle and equal distances between that angle and the bisection of the opposite sides. 28. Of all triangles on the same base and between the same parallels, the isosceles has the greatest vertical angle. 29. Of all triangles which have the same vertical angle, and whose bases pass through the same point, to determine that in which the rectangle contained by the sides is the least possible. III. 30. Shew that the perimeter of the triangle, formed by joining the feet of the perpendiculars dropped from the angles upon the opposite sides of a triangle, is less than the perimeter of any other triangle, whose angular points are on the sides of the first. 31. In an acute-angled triangle, to find a point from which if three lines be drawn to the three angles, the sum of these lines shall be the least possible. 32. Draw through the angles of an equilateral triangle three straight lines which shall form by their intersection another equilateral triangle; shew that there may be an infinite number of such triangles, and describe the greatest. 33. The sides of a triangle are divided in the same ratio; shew that the sum of the squares on the lines that join the points of division with each other, is greatest when the sides are bisected. 34. Inscribe a square in a given square. Shew that an infinite number can be inscribed, and find the least. 35. In a given rectangle ABCD inscribe another rectangle which shall have one angle at a given point P in AB; giving all the solutions and the limits of the possibility of the problem. 36. Determine which is the greatest, and which the least, of the three squares which may be inscribed in a given triangle. 37. Through the angular points of a quadrilateral, lines are drawn forming a rectangle; shew that the difference of the areas of the greatest and least rectangles which can be so formed, is equal to twice the area of the quadrilateral. IV. 38. Through a point in a circle which is not the center, to draw the least chord. 39. From two given points draw two straight lines to the same point in a given straight line, so as to include a given angle; and find the limit which the given angle must not exceed. 40. Of all straight lines which can be drawn from two given points to meet in the convex circumference of a given circle, the sum of those two will be the least, which make equal angles with the tangent at the point of concourse. 41. If any point P be taken in the plane of a circle, and PA, PB, PC,... be drawn to any number of points A, B, C,... situated symmetrically in the circumference, the sum of PA, PB,... is least when P is the center of the circle. 42. From a given point within a circle, to draw a line to the circle, so that the angle which it makes with the tangent at the point of contact may be the least possible. 43. Find a point in the circumference of a given circle, the sum of whose distances from two given lines at right angles to each other is the greatest or least possible. 44. Of all straight lines which can be drawn through a given point within a circle, find that which cuts the circumference in the greatest and least angles. 45. Let N be any point in the diameter of a circle, whose center is S, PNQ a chord drawn through N, and join SP; shew geometrically that PQ is a minimum, and the angle SPQ a maximum when PQ is perpendicular to the diameter. 46. Determine that point in the arc of a quadrant from which two lines being drawn, one to the center, and the other bisecting the radius, the included angle shall be the greatest possible. 47. A flag-staff of a given height is erected on a tower whose height is also given: at what point on the horizon will the flag-staff appear under the greatest possible angle? 48. A and B are two points within a circle; find the point P on. the circumference such that, if PAH, PBK be drawn meeting the circle in H and K, the chord HK shall be the greatest possible. 49. Find the point in a given straight line at which the tangents to a given circle will contain the greatest angle. 50. If from a point without a circle two tangents be drawn to the circle, and lines be drawn terminated by these two tangents and themselves touching the circle, shew that the least of these straight lines is that which falls between the circle and the intersection of the tangents, and makes equal angles with the two tangents, 51. Draw through a given point in the diameter of a circle a chord, which shall form with the lines joining its extremities with either extremity of the diameter, the greatest possible triangle. 52. From a given point A without a circle whose center is 0, draw a straight line cutting the circle in the points B and C, such that the area BOC may be a maximum, 53. The circumference of one circle is wholly within that of another. Find the greatest and the least straight lines that can be drawn touching the former and terminated by the latter. 54. Let one circle touch another internally, and let straight lines touch the inner circle, and be terminated by the outer; shew that the greatest of these lines is the one parallel to the common tangent at the point of contact. 55. Find the longest and shortest lines which can be drawn joining the circumferences of two circles. 56. Find the points from which two unequal circles subtend equal angles. Find also the positions of the points when the equal angles are the greatest and least possible. 57. Through either of the points of intersection of two given intersecting circles, draw the greatest possible line terminated both ways by the two circumferences. 58. Find the radii of the greatest and least circles which can be described touching two given circles neither concentric nor intersecting. 59. Two circles can be described, each of which shall touch a given circle, and pass through two given points outside the circle; shew that the angles which the two given points subtend at the two points of contact, are one greater and the other less than that which they subtend at any other point in the given circle. 60. The centers of three circles are equidistant from each other. Describe an equilateral triangle of given magnitude such that the three circles shall touch its three sides respectively on the part external to the triangle. What are the greatest and least magnitudes of the triangle that this may be possible ? VI. 61. Through a given point between two indefinite straight lines not parallel to each other, to draw a straight line which shall be terminated by them, so that the rectangle contained by its segments shall be less than the rectangle contained by the segments of any other straight line drawn through the same point and terminated by the same straight lines. 62. A, B, are two given points either within or without a given circle; find in the circumference points P, so that AP2+ BP2 may be the greatest and least possible. 63. Find the least triangle which can be circumscribed about a given circle. 64. The perimeter of an equilateral triangle inscribed in a circle is greater than the perimeter of any other isosceles triangle inscribed in the same circle. 65. Prove that the greatest right-angled triangle that can be inscribed in a circle is an isosceles triangle. 66. The square is greater than any rectangle inscribed in the same circle. 67. Inscribe the greatest parallelogram in a given semicircle. 68. AB is a fixed chord in a circle. Find the position of the chord AC, such that the diagonal through 4 of the parallelogram constructed on AB and AC as sides may be a maximum. 69. Of all quadrilateral figures contained by four given straight lines, the greatest is that which is inscriptible in a circle. 70. If on a straight line there stand two polygons, without reentrant angles, one of which encloses the other, the exterior polygon has the longest perimeter. 71. Prove that of the polygons of a given number of sides, which can be inscribed in a given circle, the greatest is that which is equilateral. 72. Of all polygons formed with given sides, the greatest is that which may be inscribed in a circle. 73. Of all polygons having equal perimeters, and the same number of sides, the equilateral polygon has the greatest area. To describe a circle which shall touch a given circle in a given point, and pass through a given point in a line given in position. Analysis. Let A be the given point in the circumference of the given circle whose center is C, and B the given point in the line DE given in position. Suppose the point G to be the center of the required circle which touches the given circle in the point A, and passes through the point B in the given line DE, then GA is equal GB, & being the center of the circle: and GC passes through the point of contact A; (Euc. III. 12.) Through C draw CF parallel to AB meeting GB produced in F. Then GC is equal to GF, (Euc. I. 6.) and GA is equal to GB, therefore BF is equal to AC the radius of the given circle. from B draw BF equal to CA and meeting CF in F, Then Gis the center of the circle which passes through the given point B in the line DE, and touches the given circle whose center is C in the given point A. To describe two circles with given radii, which shall touch each other and a given straight line, one of the circles touching the given line at a given point. Analysis. Let AB be the given straight line and C the given point in it. Suppose the circles whose centers are 0, O' described with the given radii, touch each other in the point P, and the circle whose center is O touch the line AB in the given point C, and the circle whose center is O' in some other point D. |