THEOREM C. If two tangents from one point to a circle, and the chord joining the points of contact, be drawn; then any line drawn from the intersection of the tangents which cuts the circle will be harmonically divided at its intersection with the circle and its chord. LET the tangents DE, DF, drawn from D, touch the circle A in E and F, and let EF be joined: then any line DA cutting the circle in A and B and the chord in C will be harmo nically divided, such that AC: CB :: AD: DB. E For DC2 + EC.CF = DE2 (th. 39) = AD. DB (th. 61, cor. 1). Whence DC2 AD.DBEC.CF = AD. DB — AC. CB. Hence (conv. of note 6, p. 341) the line AB is harmonically divided in C and D. THEOREM CI. If each of the angles at the base of an isosceles triangle be double of the vertical angle, the base is the greater segment of the side divided in extreme and mean ratio; and if the base of an isosceles triangle be the greater segment of the side divided in extreme and mean ratio, each of the angles at the base is double of the vertical angle. 1. LET the angles BAC, BCA, in the isosceles triangle ABC be each double of the third, ABC; then AC will be the greater segment of AB or BC divided in extreme and mean ratio. For, draw AD bisecting the angle BAC; and since BAC is double of ABC, its half, BAD, is equal to ABD or DAC: but ADC is equal to BAD and ABD; that is, to BAC or BCA, and likewise AD, DB, are equal, since the angles BAD, ABD, are equal. Hence the triangle DAC is similar A to ABC, and AC, AD, equal: and therefore AB : AC :: AC : CD; or since AD, AC, BD, are all equal, CB : BD :: BD : DC. 2. Let the base AC of the isosceles triangle ABC be the greater segment of the side BC divided in extreme and mean ratio; then each of the angles BAC, BCA, will be double of ABC. : For make BD equal to AC, and join AD; and since CB : BD :: BD : DC, we have AB AC :: AC: CD, and the angles BAC, ACD, equal, and therefore the triangles BAC, ACD, similar. Whence, since the sides AB, BC, are equal, the sides AC, AD, are also equal, and the angles ACD, ADC, also equal. Now by construction, BD is equal to AC; and therefore, also, to AD, or ADB is isosceles; and the angle ADC being equal to ABD and BAD, is double of one of them ABD: but ADC is equal to the angle DCA, and therefore to each of the angles BAC, BCA, of the triangle ABC; and hence each of these angles is double of the angle ABC at the vertex. 343 MISCELLANEOUS EXERCISES IN PLANE GEOMETRY*. 1. FROM two given points on the same side of a line given in position, to draw two lines which shall meet in that line, and make equal angles with it. 2. 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 shall be in the same straight line. 3. If a line touching two circles cut another line joining their centres, the segments of the latter will be to each other as the diameters of the circles. 4. If a straight line touch the interior of two concentric circles, and be placed in the outer, it will be bisected at the point of contact. 5. If from the extremities of the diameter of a semicircle perpendiculars be let fall on any line cutting the semicircle, the parts intercepted between those perpendiculars and the circumference are equal. 6. If on each side of any point in a circle any number of equal arcs be taken, and the extremities of each pair joined; the sum of the chords so drawn will be equal to the last chord produced to meet a line drawn from the given point through the extremity of the first arc. 7. If two circles touch each other externally or internally, any straight line drawn through the point of contact will cut off similar segments. 8. If two circles touch each other externally or internally, two straight lines drawn through the point of contact will intercept arcs, the chords of which are parallel. 9. If two circles touch each other, and also touch a straight line; the part of the line between the points of contact is a mean proportional between the diameters of the circles. 10. If a common tangent be drawn to any number of circles which touch each other internally, and from any point in this tangent as a centre a circle be described cutting the others, and from this centre lines be drawn through the intersections of the circles respectively; the segments of them within each circle will be equal. 11. If the radius of a circle be a mean proportional to two distances from the centre in the same straight line, the lines drawn from their extremities to any point in the circumference will have the same ratio that the distances of these points from the circumference have. 12. In a circle to place a straight line of a given length, so that perpendiculars drawn to it from two given points in the circumference may have a given ratio. 13. If any two chords be drawn in a circle, to intersect at right angles, then will the squares upon the four segments of those chords be together equal to the square upon the diameter of the circle. 14. If the tangents drawn to every two of three unequal circles be produced till they meet, the points of intersection will be in a straight line. * It is not expected that the student should go through all these exercises in his first study of geometry; but that the tutor should select from them fewer or more according to the capacity and talent of his pupil; requiring demonstrations of the theorems, and both constructions and demonstrations of the problems, thus selected. 15. 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. 16. The three straight lines which bisect the three angles of a triangle meet in the same point. 17. If from the angles of a triangle, lines, each equal to a given line, be drawn to the opposite sides (produced if necessary); and from any point within, lines be drawn parallel to these, and meeting the sides of the triangle; these lines will together be equal to the given line. 18. The two triangles, formed by drawing straight lines from any point within a parallelogram to the extremities of two opposite sides, are together half of the parallelogram. 19. If in the sides of a square, at equal distances from the four angles, four other points be taken, one in each side; the figure contained by the straight lines which join them shall also be a square. 20. Determine the figure formed by joining the points of bisection of the sides of a trapezium, and its ratio to the trapezium. 21. Determine the figure formed by joining the points where the diagonals of the trapezium cut the parallelogram (in the last problem), and its ratio to the trapezium. 22. If the sides of any pentagon be produced to meet, the angles formed by these lines are together equal to two right angles. 23. If the sides of any hexagon be produced to meet, the angles formed by these lines are together equal to four right angles. 24. If squares be described on the three sides of a right-angled triangle, and the extremities of the adjacent sides be joined; the triangles so formed are equal to the given triangle and to each other. 25. If from the angular points of the squares described upon the sides of a right-angled triangle, perpendiculars be let fall upon the hypothenuse produced, they will cut off equal segments; and the perpendiculars will together be equal to the hypothenuse. 26. If squares be described on the hypothenuse and sides of a right-angled triangle, and the extremities of the sides of the former and the adjacent sides of the others be joined; the sum of the squares of the lines joining them will be equal to five times the square of the hypothenuse. 27. If through any point within a triangle lines be drawn from the angles to cut the opposite sides, the segments of any one side will be to each other in the ratio compounded of the ratios of the segments of the other sides. 28. If a line be drawn from the vertex to any point in the base of a triangle, the sum of the two solids under the squares of the two sides and the alternate segments of the base will be equal to the solid under the whole base and its two segments, together with the solid under the same base and the square of the dividing line. 29. Determine a point in a line given in position, to which lines drawn from two given points may have the greatest difference possible. 30. Divide a given triangle into any number of parts, having a given ratio to each other, by lines drawn parallel to one of the sides of the triangle. 31. Through a given point between two straight lines containing a given angle, to draw a line which shall cut off a triangle equal to a given figure. 32. Divide a circle into any number of concentric equal annuli. 33. Divide it into annuli which shall have a given ratio. 34. In any quadrilateral figure circumscribing a circle, the opposite sides are equal to half the perimeter. 35. Inscribe a square in a given right-angled isosceles triangle. 36. Inscribe a square in a given quadrant of a circle. 37. Inscribe a square in a given semicircle. 38. Inscribe a square in a given segment of a circle. 39. Having given the distance of the centres of two equal circles which cut each other, inscribe a square in the space included between the two circumferences. 40. In a given segment of a circle inscribe a rectangular parallelogram whose sides shall have a given ratio. 41. In a given triangle inscribe a triangle similar to a given triangle. 42. In a given equilateral and equiangular pentagon inscribe a square. 43. In a given triangle inscribe a rhombus, one of whose angles shall be in a given point in the side of the triangle. 44. Inscribe a circle in a given quadrant. 45. 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 proportion of two to three. 46. If through the middle point of any chord of a circle two chords be drawn, the lines joining their extremities will intersect the first chord at equal distances from the middle point. 47. If in a right-angled triangle a perpendicular be drawn from the right angle to the hypothenuse, and circles inscribed within the triangles on each side of it, their diameters will be to each other as the subtending sides of the rightangled triangle. 48. If in a right-angled triangle a perpendicular be drawn from the right angle to the hypothenuse, and circles inscribed within the triangles on each side of it, they will be to each other as the segments of the hypothenuses made by the perpendicular. 49. In any triangle, if perpendiculars be drawn from the angles to the opposite sides, they will all meet in a point. 50. Three equal circles touch each other; compare the area of the triangle formed by joining their centres with the area of the triangle formed by joining the points of contact. 51. If a four-sided rectilinear figure be described about a circle, the angles subtended at the centre of the circle, by any two opposite sides of the figure, are together equal to two right angles. 52. If two given straight lines touch a circle, and if any number of other tangents be drawn, all on the same side of the centre, and all terminated by the two given tangents, the angles which they subtend at the centre of the circle shall be equal to one another. 53. If two circles cut each other, and from any point in the prolongation of the straight line which joins their intersections, two tangents be drawn, one to each circle, they shall be equal to each other. 54. To cut off from a given parallelogram a similar parallelogram which shall be any given part of it. 55. If there be any right-lined hexagonal figure, and two contiguous sides be in succession equal and parallel to two other contiguous and opposite sides, each to each; then, first, the two remaining sides will be respectively equal and parallel; secondly, the opposite angles (viz. the first and fourth, second and fifth, third and sixth,) will be equal to one another; thirdly, any diagonal joining two of those opposite angles, will divide the figure into two equal parts. 56. In any pentagonal right-lined figure, thrice the sum of the squares of the sides will be equal to the sum of the squares of the diagonals, together with four times the sum of the squares of the five right lines joining, in order, the middle points of those diagonals. 57. If there be any rectilinear figure having an even number of sides in a circle, the sum of all the angles of those angles of the figure, beginning at any one, which succeed one another, according to the odd numbers, will be equal to the sum of all the angles which succeed one another according to the even numbers. 58. If each side of any rectilinear figure, whose sides are even in number, touch a circle, the sum of the first, third, fifth, &c., beginning at any one side, and proceeding in order according to the odd numbers, will be equal to the sum of the remaining second, fourth, sixth, &c., sides, proceeding according to the even numbers. 59. If two circles intersect one another, and any right line be drawn cutting the circles, it will be proportionally divided by the circumferences of the circles. 60. Given the perpendicular drawn from the vertical angle to the base, and the difference between each side and the adjacent segment of the base made by the perpendicular; to construct the triangle. 61. Given the vertical angle, the perpendicular drawn from it to the base, and the ratio of the segments of the base made by it; to construct the triangle. 62. Given the vertical angle, the difference of the two sides containing it, and the difference of the segments of the base made by a perpendicular from the vertex; to construct the triangle. 63. Given the lengths of three lines drawn from the angles to the points of bisection of the opposite sides; to construct the triangle. 64. The sum of the descending infinite series a + b + a2 α b' b2 b3 + .... a a2 + is well b and a. De or a third proportional to a known to be expressed by 65. Every equilateral polygon circumscribed about a circle, or inscribed in a circle, is equiangular; and every equiangular polygon so circumscribed or inscribed is equilateral. 66. If from the points of contact of a regular circumscribed polygon, lines be drawn from each point of contact to its adjacent ones, the polygon so described will be regular; and if to the circle at the angular points of a regular inscribed polygon, tangents be drawn, these, by their successive adjacent intersections, will form a regular circumscribed polygon. 67. Every regular polygon is capable of inscription and circumscription by circles. 68. If in a regular inscribed polygon of an odd number of sides, parallels to each side be drawn through the angles opposite to those sides respectively, they will form by their intersections a regular circumscribed polygon. 69. If in a regular inscribed polygon of an even number of sides, lines be drawn parallel to those which join every two adjacent sides through the angle, most distant from these lines, the lines so drawn will be tangents to the circle, and their assemblage will constitute a regular circumscribed polygon. 70. If straight lines be drawn through any point to cut a circle, and the fourth harmonical points in each of them, (the given point and the intersections with the circle being the other three,) be found: all these fourth points will be in one straight line. |