ELEMENTS OF GEOMETRY. BOOK III. DEFINITIONS. A. THE radius of a circle is the straight line drawn from Book III. the centre to the circumference. I. A straight line is said to touch a circle, when it meets the circle, and, being produced, does not cut it. II. Circles are said to touch one another, which meet, but do not cut one another. III. 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. IV. And the straight line on which the greater perpendicular falls is said to be farther from the centre. PROP. I. PROB. TO find the centre of a given circle. Let ABC be the given circle; it is required to find its centre. Book III. Draw within it any straight line AB, and bisecta it in D; a 10. 1. from the point D drawb DC at right angles to AB, and pro- b 11. 1. duce 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 possible, G be the centre, and join GA, GD, GB; then, because DA is equal to DB, and DG common to the two triangles ADG, BDG the two sides AD, DG are equal to the two BD, DG, each to each; and the base GA is equal to the base GB, because they are radii of the same circle ; therefore the angle ADG is equals to the angle GDB; therefore the angle GDB is a right angled. But FDB is likewise a right angle; wherefore the angle FDB is equal to the angle GDB, the greater to the less, which is impossible. Therefore G is not the centre of the circle ABC. c 8. 1. d 7. Def. 1. B E In the same manner it can be shown that no other point but COR. From this it is manifest that if in a circle a straight line bisect another at right angles, the centre of the circle is in the line which bisects the other. PROP. II. THEOR. IF any two points be taken in the circumference of a circle, the straight line which joins them will fall within the circle. Book III. a 1. 3. b 5. 1. c 16. 1. d 19. 1. Let ABC be a circle, and A, B any two points in the cir- Take any point in AB, as E; finda D E B the triangle DAE, is produced to B, the angle DEB is PROP. III. THEOR. IF a straight line drawn through the centre of a circle bisect a straight line in the circle, which does not pass through the centre, it will cut that line at right angles; and if it cut it at right angles, it will bisect it. a 1. 3. Let ABC be a circle, and let CD, a straight line drawn through the centre, bisect any straight line AB, which does not pass through the centre, in the point F; it cuts it also at right angles. Takea E the centre of the circle, and join EA, EB. Then, because AF is equal to FB, and FE common to the two b 8. 1. triangles AFE, BFE, there are two sides in the one equal to Book III. two sides in the other; but the base EA is equal to the base EB; therefore the angle AFE is equal to the angle BFE; therefore each of the angles AFE, BFE is a right angles; wherefore the straight line CD, drawn through the centre, bisecting another, AB, which does not pass through the centre, cuts it at right. angles. B D Again, let CD cut AB at right angles; CD also bisects AB, that is AF is equal to FB. c 7. Def. 1. The same construction being made, because the radii EA, EB are equal to each other, the angle EAF is equal to d 5. 1. the angle EBF; and the right angle AFE is equal to the right angle BFE; therefore, in the two triangles EAF, EBF, there are two angles in one equal to two angles in the other; and the side EF, which is opposite to one of the equal angles in each, is common to both; therefore the other sides are equale; AF therefore is equal to FB. Wherefore, if a e 26. 1, straight line, &c. Q. E. D. PROP. IV. THEOR. IF in a circle two straight lines cut each other, which do not both pass through the centre, they do not bisect each other. Let ABCD be a circle, and AC, BD two straight lines in it, which cut each other in the point E, and do not both pass through the centre; AC, BD do not bisect each other. For, if it is possible, let AE be equal to EC, and BE to ED. If one of the lines pass through the centre, it is plain that it cannot be bisected by the other, which does not pass through the centre. But A if neither of them pass through the K E F D a 1. 3. |