Prop. xxxI. This proposition is the general case of Prop. 47, Book 1, for any similar rectilineal figure described on the sides of a right-angled triangle. The demonstration, however, here given is wholly independent of Euc. 1. 47. Prop. XXXIII. In the demonstration of this important proposition, angles greater than two right angles are employed, in accordance with the criterion of proportionality laid down in Euc. v. def. 5. This proposition forms the basis of the assumption of ares of circles for the measures of angles at their centers. One magnitude may be assumed as the measure of another magnitude of a different kind, when the two are so connected, that any variation in them takes place simultaneously, and in the same direct proportion. This being the case with angles at the center of a circle, and the arcs subtended by them; the arcs of circles can be assumed as the measures of the angles they subtend at the center of the circle. Prop. B. The converse of this proposition does not hold good when the triangle is isosceles. QUESTIONS ON BOOK VI. 1. DISTINGUISH between similar figures and equal figures. 2. What is the distinction between homologous sides, and equal sides in Geometrical figures? 3. What is the number of conditions requisite to determine similarity of figures? Is the number of conditions in Euclid's definition of similar figures greater than what is necessary? Propose a definition of similar figures which includes no superfluous condition. 4. Explain how Euclid makes use of the definition of proportion in Euc. vi. 1. 5. Prove that triangles on the same base are to one another as their altitudes. 6. If two triangles of the same altitude have their bases unequal, and if one of them be divided into m equal parts, and if the other contain of those parts; prove that the triangles have the same numerical relation as their bases. Why is this Proposition less general than Eue. vi. 1? 7. Are triangles which have one angle of one equal to one angle of another, and the sides about two other angles proportionals, necessarily similar? 8. What are the conditions, considered by Euclid, under which two triangles are similar to each other? 9. Apply Euc. vI. 2, to trisect the diagonal of a parallelogram. 10. When are three lines said to be in harmonical proportion? If both the interior and exterior angles at the vertex of a triangle (Euc vi. 3, A.) be bisected by lines which meet the base, and the base produced, in D, G; the segments BG, GD, GC of the base shall be in Harmonical proportion. 11. If the angles at the base of the triangle in the figure Euc. vi. A, be equal to each other, how is the proposition modified? 12. Under what circumstances will the bisecting line in the fig. Euc. VI. A, meet the base on the side of the angle bisected? Shew that there is an indeterminate case. 13. State some of the uses to which Euc. vi. 4, may be applied. 14. Apply Euc. vi. 4, to prove that the rectangle contained by the segments of any chord passing through a given point within a circle is constant. 15. Point out clearly the difference in the proofs of the two latter cases in Euc. vi. 7. 16. From the corollary of Euc. vr. 8, deduce a proof of Euc. 1. 47. 17. Shew how the last two properties stated in Euc. vi. 8. Cor. may be deduced from Euc. 1. 47; II. 2; vI. 17. 18. Given the nth part of a straight line, find by a Geometrical construction, the (n + 1)th part. 19. Define what is meant by a mean proportional between two given lines and find a mean proportional between the lines whose lengths are 4 and 9 units respectively. Is the method you employ suggested by any Propositions in any of the first four books? 20. Determine a third proportional to two lines of 5 and 7 units: and a fourth proportional to three lines of 5, 7, 9, units. 21. Find a straight line which shall have to a given straight line, the ratio of 1 to 5. 22. Define reciprocal figures. Enunciate the propositions proved respecting such figures in the Sixth Book. 23. Give the corollary, Euc. vI. 8, and prove thence that the Arithmetic mean is greater than the Geometric between the same extremes. 24. If two equal triangles have two angles together equal to two right angles, the sides about those angles are reciprocally proportional. 25. Give Algebraical proofs of Prop. 16 and 17 of Book vi. 26. Enunciate and prove the converse of Euc. vi. 15. 27. Explain what is meant by saying, that "similar triangles are in the duplicate ratio of their homologous sides." 28. What are the data which determine triangles both in species and magnitude? How are those data expressed in Geometry? 29. If the ratio of the homologous sides of two triangles be as 1 to 4, what is the ratio of the triangles? And if the ratio of the triangles be as 1 to 4, what is the ratio of the homologous sides? 30. Shew that one of the triangles in the figure, Euc. Iv. 10, is a mean proportional between the other two. 31. What is the algebraical interpretation of Euc. vi. 19? 32. From your definition of Proportion, prove that the diagonals of a square are in the same proportion as their sides. 33. What propositions does Euclid prove respecting similar polygons? 34. The parallelograms about the diameter of a parallelogram are similar to the whole and to one another. Shew when they are equal. 35. Prove Algebraically, that the areas (1) of similar triangles and (2) of similar parallelograms are proportional to the squares of their homologous sides. 36. How is it shewn that equiangular parallelograms have to one another the ratio which is compounded of the ratios of their bases and altitudes ? 37. To find two lines which shall have to each other, the ratio compounded of the ratios of the lines A to B, and C to D. 38. State the force of the condition "similarly described;" and shew that, on a given straight line, there may be described as many polygons of different magnitudes, similar to a given polygon, as there are sides of different lengths in the polygon. 39. Describe a triangle similar to a given triangle, and having its area double that of the given triangle. 40. The three sides of a triangle are 7, 8, 9 units respectively; determine the length of the lines which meeting the base, and the base produced, bisect the interior angle opposite to the greatest side of the triangle, and the adjacent exterior angle. 41. The three sides of a triangle are 3, 4, 5 inches respectively; find the lengths of the external segments of the sides determined by the lines which bisect the exterior angles of the triangle. 42. What are the segments into which the hypotenuse of a rightangled triangle is divided by a perpendicular drawn from the right angle, if the sides containing it are a and 3a units respectively? 43. If the three sides of a triangle be 3, 4, 5 units respectively: what are the parts into which they are divided by the lines which bisect the angles opposite to them? 44. If the homologous sides of two triangles be as 3 to 4, and the area of one triangle be known to contain 100 square units; how many square units are contained in the area of the other triangle? 45. Prove that if BD be taken in AB produced (fig. Euc. vi. 30) equal to the greater segment AC, then AD is divided in extreme and mean ratio in the point B. Shew also, that in the series 1, 1, 2, 3, 5, 8, &c. in which each term is the sum of the two preceding terms, the last two terms perpetually approach to the proportion of the segments of a line divided in extreme and mean ratio. Find a general expression (free from surds) for the nth term of this series. 46. The parts of a line divided in extreme and mean ratio are incommensurable with each other. 47. Shew that in Euclid's figure (Euc. I. 11.) four other lines, besides the given line, are divided in the required manner. 48. Enunciate Euc. vi. 31. What theorem of a previous book is included in this proposition? 49. What is the superior limit, as to magnitude, of the angle at the circumference in Euc. vi. 33? Shew that the proof may be extended by withdrawing the usually supposed restriction as to angular magnitude; and then deduce, as a corollary, the proposition respecting the magnitudes of angles in segments greater than, equal to, or less than a semicircle. 50. The sides of a triangle inscribed in a circle are a, b, c, units respectively: find by Euc. vi. c, the radius of the circumscribing circle. 51. Enunciate the converse of Euc. vi. D. 52. Shew independently that Euc. VI. D, is true when the quadrilateral figure is rectangular. 53. Shew that the rectangles contained by the opposite sides of a quadrilateral figure which does not admit of having a circle described about it, are together greater than the rectangle contained by the diagonals. 54. What different conditions may be stated as essential to the possibility of the inscription and circumscription of a circle in and about a quadrilateral figure? 55. Point out those propositions in the Sixth Book in which Euclid's definition of proportion is directly applied. 56. Explain briefly the advantages gained by the application of analysis to the solution of Geometrical Problems. 57. In what cases are triangles proved to be equal in Euclid, and in what cases are they proved to be similar? To inscribe a square in a given triangle. Analysis. Let ABC be the given triangle, of which the base B and the perpendicular AD are given. Let FGHK be the required inscribed square. therefore GF is to GB, as AD is to AB. Let BF be joined and produced to meet a line drawn from A parallel to the base BC in the point E. Synthesis. Then the triangles BGF, BAE are similar, and AE is to AB, as GF is to GB, but GF is to GB, as AD is to AB; wherefore AE is to AB, as AD is to AB; hence AE is equal to AD. Through the vertex A, draw AE parallel to BC the base of the triangle, make AE equal to AD, join EB cutting AC in F, through F, draw FG parallel to BC, and FK parallel to AD; also through G draw GH parallel to AD. Then GHKF is the square required. The different cases may be considered when the triangle is equilateral, scalene, or isosceles, and when each side is taken as the base. PROPOSITION II. THEOREM. If from the extremities of any diameter of a given circle, perpendiculars be drawn to any chord of the circle, they shall meet the chord, or the chord produced in two points which are equidistant from the center. First, let the chord CD intersect the diameter AB in Z, but not at right angles; and from A, B, let AE, BF be drawn perpendicular to CD. Then the points F, E are equidistant from the center of the chord CD. Join EB, and from I the center of the circle, draw IG perpendicular to CD, and produce it to meet EB in H. Then IG bisects CD in G; (III. 2.) and IG, AE being both perpendicular to CD, are parallel. (1. 29.) Therefore BI is to BH, as IA is to HE; (VI. 2.) and BH is to FG, as HE is to GE; therefore BI is to FG, as IA is to GE; but BI is equal to IA; therefore FG is equal to GE. It is also manifest that DE is equal to CF. When the chord does not intersect the diameter, the perpendiculars intersect the chord produced. PROPOSITION III. THEOREM. If two diagonals of a regular pentagon be drawn to cut one another, the greater segments will be equal to the side of the pentagon, and the diagonals will cut one another in extreme and mean ratio. Let the diagonals AC, BE be drawn from the extremities of the side AB of the regular pentagon ABCDE, and intersect each other in the point H. Then BE and AC are cut in extreme and mean ratio in H, and the greater segment of each is equal to the side of the pentagon. Let the circle ABCDE be described about the pentagon. (IV. 14.) Because EA, AB are equal to AB, BC, and they contain equal angles; therefore the base EB is equal to the base AC, (1. 4.) and the triangle EAB is equal to the triangle CBA, and the remaining angles will be equal to the remaining angles, each to each, to which the equal sides are opposite. Therefore the angle BAC is equal to the angle ABE; and the angle AHE is double of the angle BAH, (1. 32.) but the angle EAC is also double of the angle BAC, (vI. 33.) therefore the angle HAE is equal to AHE, and consequently HF is equal to EA, (1. 6.) or to AB. the angle ABE is equal to the angle AEB; |