Book II. PROP. XIV. PROB. a 45. 1. b 5.2. c 47. 1. TO describe a square that shall be equal to a given rectilineal figure. Let A be the given rectilineal figure; it is required to describe a square that shall be equal to A. A H Describe the rectangular parallelogram BCDE, equal to the rectilineal figure A. If then the sides of it BE, ED be equal to each other, it is a square, and what was required is done; but if they be not equal, produce one of them BE to F, and make EF equal to ED, and bisect BF in G; and from the centre G, at the distance GB, or GF, describe the semicircle BHF, and produce DE to H, and join GH.— Then, because the A B D straight line BF is divided into two equal parts in the point G, and into two unequal in the point E, the rectangle BE.EF together with the square of EG are equal to the square of GF, or GH. But the squares of HE and EG are equal to the square of GH; therefore BE.EF+EG2=HE+EG®. Take away the square of EG, which is common to both, and the remaining rectangle BE.EF is equal to the square of EH. But BD is the rectangle contained by BE and EF, because EF is equal to ED; therefore BD is equal to the square of EH; and BD is also equal to the rectilineal figure A; therefore the rectilineal figure A is equal to the square of EH. Wherefore a square has been made equal to the given rectilineal figure A, viz. the square described upon EH. Which was to be done. Book II. PROP. A. THEOR. IF one side of a triangle be bisected, the sum of See N. the squares of the other two sides is double of the square of half the side bisected, and of the square of the line drawn from the point of bisection to the opposite angle of the triangle. Let ABC be a triangle, of which the side BC is bisected in D, and DA drawn to the opposite angle; the squares of BA and AC are together double of the squares of BD and DA. From A draw AE perpendicular to BC; then AB2=a BE2+AE2, and AC2=CE2+ AE2; wherefore AB2+AC2— BE2+CE2+2AE2. But, because the line BC is cut equally in D, and unequally in E, BE2+CE2-b2BD2+2DE2; therefore AB+ AC2=2BD2 B +2DE2+2AE2. Now DE2+AE2-a AD2; and 2DE2+ 2AE2=2AD2; wherefore AB2+AC2=2BD2+2AD2. Therefore, &c. Q. E. D. b 9.2. I Book II. a 15. 1. b 29. 1. c 34. 1. d 26. 1. e A. 2. PROP. B. THEOR. THE sum of the squares of the diameters of any parallelogram is equal to the sum of the squares of the sides of the parallelogram. Let ABCD be a parallelogram, of which the diameters are AC and BD; the sum of the squares of AC and BD is equal to the sum of the squares of AB, BC, CD, DA. Let AC and BD intersect each other in E. Because the vertical angles AED, CEB are equal, and also the alternate angles EAD, ECB, the triangles ADE, CEB have two angles in the one equal to two angles in the other, each to each: but the sides AD and BC, which are opposite to equal angles in these triangles are also equal; therefore the other sides which are opposite to the equal angles are also equald, viz. AE to EC, and ED to EB. Since, therefore, BD is bisected in E, e = B A E AB2+AD2= 2BE2+2AE2; and for the same reason CD2+ BC2 = 2BE2 + 2EC2 2BE2 + 2AE2, because EC=AE. Therefore AB + AD2 + DC2 + BC2 = 4BE2 + 4AE2. But f2.Cor.8.2. 4BE2-BD2, and 4AE2-AC2, because BD and AC are both bisected in E; therefore AB2+ AD2+CD2+BC2 = BD2+AC2. Therefore, the sum of the squares, &c. Q. E. D. COR. From this demonstration it is manifest that the diameters of every parallelogram bisect each other. PROP. C. THEOR. IF a perpendicular be drawn from any angle of a triangle to the opposite side, the difference of the squares of the sides is equal to the difference of the squares of the segments of the base. Let ABC be a triangle, having the side AB greater than AC, and AD a perpendicular from the angle A to the opposite side BC; AB AC-BD2_DC". AB-AC2=BD2-DC2, by taking equals from equals. Therefore, if a perpendicular, &c. Q. E. D. Book II. a 47. 1. Book II. a C. 2. PROP. D. THEOR. IF a perpendicular be drawn from any angle of a triangle to the opposite side, the rectangle contained by the sum and difference of the sides is equal to the rectangle contained by the sum and difference of the segments of the base, that is, to the rectangle contained by the base, and the difference or sum of the distances intercepted between the extremities of the base and the perpendicular, according as the perpendicular falls within or without the triangle. Let ABC be a triangle*, having the side AB greater than AC, and AD a perpendicular from the angle A to the opposite side BC; then AB+AC. AB-AC=BD+DC.BD—DC. AB2-AC2-BD2-DC2, a b Cor. 5.2. But and AB2-AC-AB+AC.AB-ACb, BD2-DC? BD+DC.BD-DC; therefore AB+AC,AB-AC÷BD+DC.BD—DC. Now when the perpendicular AD falls within the triangle, BD+DC is equal to the base BC, and BD-DC is equal to the difference of the segments of the base; but when AD falls without the triangle, BD-DC is equal to the base BC, and BD+DC is equal to the sum of the segments of the base. Therefore, if a perpendicular, &c. Q. E. D. * See the last figures. |