BOOK H. PROP. A. THEOR. IN See N. N any triangle, the difference between the fquares of two of the fides is equal to twice the rectangle contained by the bafe, and the fegment of it intercepted between the perpendicular drawn to it from the oppofite angle, and the point in which it a 1. Y. b 12. 1. C 3. I. is bifected. Let ABC be a triangle, of which the fide AB is greater than AC; and bifect the base BC in D, and draw AE perpendicular to BC; the fquare of AB is greater than the fquare of AC, by twice the rectangle BC, DE. c 1 A Produce BC to F, and make EF equal to EB: and because BF is double of BE, and BC double of BD; the remainder CF is double of the remainder DE: And because BF is divided into two equal parts in the point E, and into two unequal parts in the point C; the rectangle BC. CF, together with the 4.5. 2. fquare of CE, is equal to the fquare B of BE add to thefe equals the e 1. 2. e DE C R fquare of EA; and the rectangle BC, CF, together with the fquares of CE, EA, is equal to the squares of BE, EA: But the rectangle BC, CF is double of the rectangle BC, DE, bef 47. 1. caufe CF is double of DE; and the fquare of AC is equal f to the fquares of CE, EA, because CEA is a right angle; and the fquare of AB is equal to the squares of BE, EA; therefore twice the rectangle BC, DE together with the fquare of AC, is equal to the fquare of AB; that is, twice the rectangle BC, DE is equal to the excefs of the fquare of AB above the fquare of AC. Wherefore, &c. Q. E. D. g 4. I COR. And if AF be joined, the triangle ABF is ifofceles ; hence it is manifeft, that if any ftraight line AC be drawn from the vertex to the bafe of an ifofceles triangle ABF; the fquare of the fide AB is equal to the fquare of the line AC, together with the rectangle BC, CF of the fegments of the base. THE PROP. B. THEOR. HE fquares of two fides of a triangle are together double of the fquare of half the bafe, and of the fquare of the ftraight line drawn from the vertex to bifect the bafe. Let Let ABC be a triangle; and let the base BC be bisected a in Book II. D, and join AD: the fquares of BA, AC together are double of the fquares of BD, DA. A a 10. I. C d 9.2. If the angles at D be right angles, the fquare of BA is equal b to the fquares of BD, DA, and the fquare of AC to the fquares b 47. I. of CD, DA, or of BD, DA; therefore the fquares of BA, AC are double of the fquares of BD, DA. But, if the angles at D be not right angles, from A draw c 12. 1. AE perpendicular to BC and becaufe BC is bifected in D, the fquares of BE, EC are doubled of the fquares of BD, DE: add to thefe equals twice the fquare of EA; and the fquares of BE, EC, together with twice the fquare of EA, are double of the fquares of BD, DE, EA: And the fquare of DA is equal to the fquares of DE, EA, because DEA is a right angle; therefore the fquares of BE, EC, together with twice the fquare of EA, are double of the fquares of BD, DA: but the fquare of BA is equal to the fquares of BE, EA, and the fquare of AC to the fquares of CE, EA; therefore the fquares of BA, AC are equal to the fquares of BE, EC, together with twice the fquare of EA; and it has been demonftrated, that the fquares of BE, EC, together with twice the fquare of EA, are double of the fquares of BD, DA; therefore the fquares of BA, AC are double of the squares of BD, DA. Wherefore, &c. Q. E. D. THE b B PROP. C. THEOR. D HE fquares of the two diameters of a parallelogram are together equal to the fquares of its four fides. Let ABCD be a parallelogram, of which the diameters are AC, BD; the fquares of AC, BD together are equal to the fquares of the four fides AB, BC, CD, DA. Let AC, BD cut one another in E; and because AC meets the parallels AD, BC, the alternate angles DAE, BCE are equal a; and the angle AED is equal to its vertical angle BEC; there- b 15. 1. fore two angles of the triangle AED are equal to two of the triangle BEC; and the fides AD, BC, oppofite to equal angles, are alfo equal; therefore their other fides are equal, each to c 26. 1. each, viz. AE to EC, and RE to ED: and becaufe AE is drawn from the vertex A of the triangle BAD, to bifect the d B. 2. . BOOK II. base BD; the fquares of BA, AD are doubled of the fquares ~ of BE, EA. For the fame reafon, the fquares of BC, CD are doubled of the fquares of BE, EC, or of BE, EA, becaufe EC is equal to EA: therefore the fquares of BA, AD, BC, CD are quadruple of the fquares of BE, EA: but the e3Cor-4.2. fquare of BD is quadruple of the fquare of BE, becaufe BD is A B D E C the four fides are quadruple of BE, EA; therefore the squares of BD, AC are equal to the fquares of the four fides AB, BC, CD, DA. Wherefore, &c. Q E. D. COR. Hence it is manifeft, that the diameters of a parallelobifect one another. gram THE THE ELEMENTS OF EUCLID. AN a BOOK III. DEFINITIONS. A. N arch of a circle is any part of the circumference. B. A chord is any ftraight line in a circle, terminated both ways by the circumference. I. Omitted. A ftraight line is said to touch a circle, when it meets the circle, and being produced, does not cut it. III. Circles are faid to touch one another, which meet, but do not cut one another. IV. Straight lines are faid to be equally di- And the ftraight line on which the VI. Book. III. 2 10. I. To find the centre of a given circle. Let ABC be the given circle; it is required to find its centre. b Draw within it any straight line AB, and bifecta it in D ; 11. 1. from the point D draw DC at right angles to AB, and produce it to E, and bisect CE in F: The point F is the centre of the circle ABC. For, if it be not, let, if poffible, G be the centre, and join GA, GD, GB: Then, becaufe DA is equal to DB, and DG common to the two triangles ADG, BDG, the two fides AD, DG are equal to the two BD, DG, each to each; and the base GA is equal to the bafe GB, because they are drawn from the centre |