b 16. 1. c 15. 1. be Book I. the same parallels DK and LO; and the square AG is also equivalent to the parallelogram AO, since they are upon the same base AB, and between the same parallels FO and ABC; therefore, the square AG is equivalent to the parallelogram AL. In like manner, by producing EC and IH, it may demonstrated, that the square BI is equivalent to the parallelogram CL; wherefore, the whole square ACED is equivalent to the two squares AG and BI. Therefore, in any right-angled triangle, &c. Q. E. D. Cor. If the square described upon one of the sides of a triangle, be equivalent to the sum of the squares described upon the other two sides, the angle contained by these two sides is a right angle. ELEMENTS OF GEOMETRY. BOOK II. DEFINITIONS. I.—A STRAIGHT line is said to touch a circle, when it Book II. meets the circle, and being produced, does not cut it. The touching line is called a tangent. II.-Circles are said to touch one another which meet, but do not cut one another. III.-An arc of a circle is any part of the circumference; and the straight line joining its extremities is called a chord. IV.-Straight lines are said to be equally distant from the centre of a circle, when the perpendiculars drawn to them from the centre are equal; and the straight line on which the greater perpendicular falls, is said to be farther from the centre. V.—A segment of a circle is the figure contained by a straight line and the arc which it cuts off. VI. An angle in a segment of a circle, is the angle contained by two straight lines, drawn from any point in the circumference of the segment, to the extremities of the straight line, which is the base of the segment. Book II. VII. And an angle is said to insist or stand upon the arch intercepted between the straight lines which contain the angle. VIII. The sector of a circle, is the figure contained by two radii, and the arc of the circumference between them. IX.-When all the angular points of one rectilineal figure are placed in the sides of another, and the one wholly within the other, the former is said to be inscribed in the latter, or the latter described about the former. X.-When all the angular points of a rectilineal figure are in the circumference of a circle, the figure is said to be inscribed in the circle, or the circle to be described about it. XI. When the sides of a rectilineal figure are tangents to a circle, the figure is said to be described about the circle, or the circle to be inscribed in it. Book II. PROP. I. THEOREM. If a straight line CD, drawn through the centre E of a circle, bisect a straight line AB in the circle, which does not pass through the centre, it will cut it at right angles; and if it cut it at right angles, it will bisect it. 14. 1. s Sup. a 4. 1. Join EA and EB; then, because EA is equal to EB, and eCor. Def. AF to FB and FE common to the two triangles, AFE and BFE, they are equal in every respect. Hence, the angle AFE is equal to the angle BFE; and, therefore, each of them is a right angle; wherefore, CD cuts AB at right angles. A C b Def. 7. E F B Ꭰ c 3. 1. Again, let CD cut AB at right angles; CD also bisects AB. The same construction being made, because the radii EA and EB are equal, the angle EAF is equal to the angle EBF; and the right angle EFA is equal to the right angle EFB, and the side EF is common to both; therefore, in the two triangles EAF and EBF, there are two angles and a side in the one equal to two angles and a corresponding side in the other; therefore, the triangles are equal in every respect; AF is therefore equal to FB. d 13. 1. Wherefore, if a straight line, &c. Q. E. D. Cor. 1. If a chord be bisected at right angles by a straight line, that straight line will pass through the centre; and, hence, if it be produced both ways to the circumference, and then bisected, the point of bisection will be the centre of the circle. Cor. 2. If two chords be bisected at right angles, the point of intersection of the bisecting lines, will be the centre of the circle. Cor. 3. Hence a circle may be described about a triangle; for if two of its sides be considered as chords, and be bisected as in Cor. 2, the point where the bisecting lines meet, will be the centre of the circle. Cor. 4. Hence, also, a circle may be completed, of which a segment is given; for the centre may be found as in the two last Cors. Book II. PROP. II. THEOREM. The diameter AD is the greatest straight line in a circle; and of all others, BC, FG; BC, which is nearer to the centre, is always greater than FG, the one more remote, and the greater is nearer to the centre than the less. From the centre E, draw EH, EK, perpendiculars to BC, a Cor.1.4.1. FG, and join EB, EC, EF; and because AE is equal to EB, and ED to EC, AD is equal to EB, EC; but EB, EC, are great b Cor. Ax.12. er than BC; wherefore, AD is also greater than BC. c 1. 2. And because BC is nearer to the centre than FG, EH is less than EK, but BC is double of BH, and FG double of FKc. Now, the square of EB is equal to the square of EF, but the square of EB is equivalent to d 18. 1. the squares of EH and HB, and the K A B E square of EF is equivalent to the squares of EK and KFd; therefore, the squares of EH, HB, are equivalent to the squares of EK, KF; of which the square of EH is less than the square of EK, consequently, the square of BH is greater than the square of FK, and the straight line BH greater than FK; and, therefore, BC is greater than FG. Next, let BC be greater than FG, BC is nearer to the centre than FG, that is, the same construction being made, EH is less than EK. Because BC is greater than FG, BH is likewise greater than KF; but the squares of BH, HE, are equivalent to the squares of FK, KE, of which the square of BH is greater than the square of FK, because BH is greater than FK; therefore, the square of EH is less than the square of EK, and the straight line EH less than FK. Wherefore, the diameter, &c. Q. E. D. Cor. Equal straight lines in a circle are equally distant from the centre; and those which are equally distant from the centre, are equally distant from one another. |